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UCLNLP

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adversarial_qa

like 33

12a9498c40e25c2661e5ac3c362059fdacee3e66

Exhibition_game

The Flying Fathers, a Canadian group of Catholic priests, regularly toured North America playing exhibition hockey games for charity. One of the organization's founders, Les Costello, was a onetime NHL player who was ordained as a priest after retiring from professional hockey. Another prominent exhibition hockey team is the Buffalo Sabres Alumni Hockey Team, which is composed almost entirely of retired NHL players, the majority of whom (as the name suggests) played at least a portion of their career for the Buffalo Sabres.

What did Les Costello do in terms of hockey, before he became a priest?

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Exhibition_game

The Flying Fathers, a Canadian group of Catholic priests, regularly toured North America playing exhibition hockey games for charity. One of the organization's founders, Les Costello, was a onetime NHL player who was ordained as a priest after retiring from professional hockey. Another prominent exhibition hockey team is the Buffalo Sabres Alumni Hockey Team, which is composed almost entirely of retired NHL players, the majority of whom (as the name suggests) played at least a portion of their career for the Buffalo Sabres.

Of the Buffalo Sabers Alumni Team, and the Flying Fathers, which one doesn't have mostly former NHL players?

000e766b8db1ee5b4f927a1a862f519be27dfa08

Exhibition_game

The Flying Fathers, a Canadian group of Catholic priests, regularly toured North America playing exhibition hockey games for charity. One of the organization's founders, Les Costello, was a onetime NHL player who was ordained as a priest after retiring from professional hockey. Another prominent exhibition hockey team is the Buffalo Sabres Alumni Hockey Team, which is composed almost entirely of retired NHL players, the majority of whom (as the name suggests) played at least a portion of their career for the Buffalo Sabres.

What distinction do the members of the BSAHT share?

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Exhibition_game

Under the 1995–2004 National Hockey League collective bargaining agreement, teams were limited to nine preseason games. From 1975 to 1991, NHL teams sometimes played exhibition games against teams from the Soviet Union in the Super Series, and in 1978, played against World Hockey Association teams also in preseason training. Like the NFL, the NHL sometimes schedules exhibition games for cities without their own NHL teams, often at a club's minor league affiliate (e.g. Carolina Hurricanes games at Time Warner Cable Arena in Charlotte, home of their AHL affiliate; Los Angeles Kings games at Citizens Business Bank Arena in Ontario, California, home of their ECHL affiliate; Montreal Canadiens games at Colisée Pepsi in Quebec City, which has no pro hockey but used to have an NHL team until 1995; Washington Capitals at 1st Mariner Arena in the Baltimore Hockey Classic; various Western Canada teams at Credit Union Centre in Saskatoon, a potential NHL expansion venue). Since the 2000s, some preseason games have been played in Europe against European teams, as part of the NHL Challenge and NHL Premiere series. In addition to the standard preseason, there also exist prospect tournaments such as the Vancouver Canucks' YoungStars tournament and the Detroit Red Wings' training camp, in which NHL teams' younger prospects face off against each other under their parent club's banner.

what other league is similar to the same type collective bargaining agreement as the NHL when it comes to games

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Exhibition_game

Under the 1995–2004 National Hockey League collective bargaining agreement, teams were limited to nine preseason games. From 1975 to 1991, NHL teams sometimes played exhibition games against teams from the Soviet Union in the Super Series, and in 1978, played against World Hockey Association teams also in preseason training. Like the NFL, the NHL sometimes schedules exhibition games for cities without their own NHL teams, often at a club's minor league affiliate (e.g. Carolina Hurricanes games at Time Warner Cable Arena in Charlotte, home of their AHL affiliate; Los Angeles Kings games at Citizens Business Bank Arena in Ontario, California, home of their ECHL affiliate; Montreal Canadiens games at Colisée Pepsi in Quebec City, which has no pro hockey but used to have an NHL team until 1995; Washington Capitals at 1st Mariner Arena in the Baltimore Hockey Classic; various Western Canada teams at Credit Union Centre in Saskatoon, a potential NHL expansion venue). Since the 2000s, some preseason games have been played in Europe against European teams, as part of the NHL Challenge and NHL Premiere series. In addition to the standard preseason, there also exist prospect tournaments such as the Vancouver Canucks' YoungStars tournament and the Detroit Red Wings' training camp, in which NHL teams' younger prospects face off against each other under their parent club's banner.

What change was instituted?

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Exhibition_game

Under the 1995–2004 National Hockey League collective bargaining agreement, teams were limited to nine preseason games. From 1975 to 1991, NHL teams sometimes played exhibition games against teams from the Soviet Union in the Super Series, and in 1978, played against World Hockey Association teams also in preseason training. Like the NFL, the NHL sometimes schedules exhibition games for cities without their own NHL teams, often at a club's minor league affiliate (e.g. Carolina Hurricanes games at Time Warner Cable Arena in Charlotte, home of their AHL affiliate; Los Angeles Kings games at Citizens Business Bank Arena in Ontario, California, home of their ECHL affiliate; Montreal Canadiens games at Colisée Pepsi in Quebec City, which has no pro hockey but used to have an NHL team until 1995; Washington Capitals at 1st Mariner Arena in the Baltimore Hockey Classic; various Western Canada teams at Credit Union Centre in Saskatoon, a potential NHL expansion venue). Since the 2000s, some preseason games have been played in Europe against European teams, as part of the NHL Challenge and NHL Premiere series. In addition to the standard preseason, there also exist prospect tournaments such as the Vancouver Canucks' YoungStars tournament and the Detroit Red Wings' training camp, in which NHL teams' younger prospects face off against each other under their parent club's banner.

What was the main focus of the article in for the NHL teams agreement

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Exhibition_game

Under the 1995–2004 National Hockey League collective bargaining agreement, teams were limited to nine preseason games. From 1975 to 1991, NHL teams sometimes played exhibition games against teams from the Soviet Union in the Super Series, and in 1978, played against World Hockey Association teams also in preseason training. Like the NFL, the NHL sometimes schedules exhibition games for cities without their own NHL teams, often at a club's minor league affiliate (e.g. Carolina Hurricanes games at Time Warner Cable Arena in Charlotte, home of their AHL affiliate; Los Angeles Kings games at Citizens Business Bank Arena in Ontario, California, home of their ECHL affiliate; Montreal Canadiens games at Colisée Pepsi in Quebec City, which has no pro hockey but used to have an NHL team until 1995; Washington Capitals at 1st Mariner Arena in the Baltimore Hockey Classic; various Western Canada teams at Credit Union Centre in Saskatoon, a potential NHL expansion venue). Since the 2000s, some preseason games have been played in Europe against European teams, as part of the NHL Challenge and NHL Premiere series. In addition to the standard preseason, there also exist prospect tournaments such as the Vancouver Canucks' YoungStars tournament and the Detroit Red Wings' training camp, in which NHL teams' younger prospects face off against each other under their parent club's banner.

What resulted from the change?

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Exhibition_game

Under the 1995–2004 National Hockey League collective bargaining agreement, teams were limited to nine preseason games. From 1975 to 1991, NHL teams sometimes played exhibition games against teams from the Soviet Union in the Super Series, and in 1978, played against World Hockey Association teams also in preseason training. Like the NFL, the NHL sometimes schedules exhibition games for cities without their own NHL teams, often at a club's minor league affiliate (e.g. Carolina Hurricanes games at Time Warner Cable Arena in Charlotte, home of their AHL affiliate; Los Angeles Kings games at Citizens Business Bank Arena in Ontario, California, home of their ECHL affiliate; Montreal Canadiens games at Colisée Pepsi in Quebec City, which has no pro hockey but used to have an NHL team until 1995; Washington Capitals at 1st Mariner Arena in the Baltimore Hockey Classic; various Western Canada teams at Credit Union Centre in Saskatoon, a potential NHL expansion venue). Since the 2000s, some preseason games have been played in Europe against European teams, as part of the NHL Challenge and NHL Premiere series. In addition to the standard preseason, there also exist prospect tournaments such as the Vancouver Canucks' YoungStars tournament and the Detroit Red Wings' training camp, in which NHL teams' younger prospects face off against each other under their parent club's banner.

What is a recent addition?

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Exhibition_game

Under the 1995–2004 National Hockey League collective bargaining agreement, teams were limited to nine preseason games. From 1975 to 1991, NHL teams sometimes played exhibition games against teams from the Soviet Union in the Super Series, and in 1978, played against World Hockey Association teams also in preseason training. Like the NFL, the NHL sometimes schedules exhibition games for cities without their own NHL teams, often at a club's minor league affiliate (e.g. Carolina Hurricanes games at Time Warner Cable Arena in Charlotte, home of their AHL affiliate; Los Angeles Kings games at Citizens Business Bank Arena in Ontario, California, home of their ECHL affiliate; Montreal Canadiens games at Colisée Pepsi in Quebec City, which has no pro hockey but used to have an NHL team until 1995; Washington Capitals at 1st Mariner Arena in the Baltimore Hockey Classic; various Western Canada teams at Credit Union Centre in Saskatoon, a potential NHL expansion venue). Since the 2000s, some preseason games have been played in Europe against European teams, as part of the NHL Challenge and NHL Premiere series. In addition to the standard preseason, there also exist prospect tournaments such as the Vancouver Canucks' YoungStars tournament and the Detroit Red Wings' training camp, in which NHL teams' younger prospects face off against each other under their parent club's banner.

What country is in the NHL besides the US

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Exhibition_game

Under the 1995–2004 National Hockey League collective bargaining agreement, teams were limited to nine preseason games. From 1975 to 1991, NHL teams sometimes played exhibition games against teams from the Soviet Union in the Super Series, and in 1978, played against World Hockey Association teams also in preseason training. Like the NFL, the NHL sometimes schedules exhibition games for cities without their own NHL teams, often at a club's minor league affiliate (e.g. Carolina Hurricanes games at Time Warner Cable Arena in Charlotte, home of their AHL affiliate; Los Angeles Kings games at Citizens Business Bank Arena in Ontario, California, home of their ECHL affiliate; Montreal Canadiens games at Colisée Pepsi in Quebec City, which has no pro hockey but used to have an NHL team until 1995; Washington Capitals at 1st Mariner Arena in the Baltimore Hockey Classic; various Western Canada teams at Credit Union Centre in Saskatoon, a potential NHL expansion venue). Since the 2000s, some preseason games have been played in Europe against European teams, as part of the NHL Challenge and NHL Premiere series. In addition to the standard preseason, there also exist prospect tournaments such as the Vancouver Canucks' YoungStars tournament and the Detroit Red Wings' training camp, in which NHL teams' younger prospects face off against each other under their parent club's banner.

Where did the inspiration come from?

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Exhibition_game

Under the 1995–2004 National Hockey League collective bargaining agreement, teams were limited to nine preseason games. From 1975 to 1991, NHL teams sometimes played exhibition games against teams from the Soviet Union in the Super Series, and in 1978, played against World Hockey Association teams also in preseason training. Like the NFL, the NHL sometimes schedules exhibition games for cities without their own NHL teams, often at a club's minor league affiliate (e.g. Carolina Hurricanes games at Time Warner Cable Arena in Charlotte, home of their AHL affiliate; Los Angeles Kings games at Citizens Business Bank Arena in Ontario, California, home of their ECHL affiliate; Montreal Canadiens games at Colisée Pepsi in Quebec City, which has no pro hockey but used to have an NHL team until 1995; Washington Capitals at 1st Mariner Arena in the Baltimore Hockey Classic; various Western Canada teams at Credit Union Centre in Saskatoon, a potential NHL expansion venue). Since the 2000s, some preseason games have been played in Europe against European teams, as part of the NHL Challenge and NHL Premiere series. In addition to the standard preseason, there also exist prospect tournaments such as the Vancouver Canucks' YoungStars tournament and the Detroit Red Wings' training camp, in which NHL teams' younger prospects face off against each other under their parent club's banner.

Where did the change originate?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

What change was brought about?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

One of the Canadian MLB teams is the Toronto Blue Jays - what is the other Canadian MLB team?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

Why did the teams not play each other regularly?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

What was a tradition in the country?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

What was the name of the longer exhibition cup?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

The team that moved to Washington DC for the 2005 season used to have a tradition of playing against what other team?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

What game did the team that used to play regular exhibition games against other local rivals from 1946 to 1983 formerly participate in?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

How did the MLB contest interleague play?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

What nationality is represented by the Yankees?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

The team that moved to Washington DC in 2005 are in what league?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

What population competed and made things more simple?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

Why don't MLB teams typically play regular exhibition games anymore?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

What are the Blue Jay and Expos a part of?

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Exhibition_game

Several MLB teams used to play regular exhibition games during the year against nearby teams in the other major league, but regular-season interleague play has made such games unnecessary. The two Canadian MLB teams, the Toronto Blue Jays of the American League and the Montreal Expos of the National League, met annually to play the Pearson Cup exhibition game; this tradition ended when the Expos moved to Washington DC for the 2005 season. Similarly, the New York Yankees played in the Mayor's Trophy Game against various local rivals from 1946 to 1983.

What kind of league played the Pearson Cup?

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Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

What is the complement of two hundred seventy degrees?

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Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

In the equation, what was the lowercase g supposed to indicate?

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Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

What do you need information about to understand the entirety of a group?

474e04a6f40b2e4c87ca91d1f0fe23434a5300b0

Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

In the equation, what was the notation used to indicate a subset?

c3f4c4b4a1fadba931bb6f5bec6bc417abbb48f6

Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

For what measurement is its complement the same number?

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Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

What is the inverse for 90?

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Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

What information does the subgroup give?

8f69f543c0803a28118847a9841c66a81f4e2153

Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

What makes a subgroup necessary?

b32511b4795c0189c93f41d5f09535988da1fa16

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Under what theory is the relationship between the structure of groups and the structure of fields being studied?

b010bf7a6ccd080e17c0d5a24fe849fc872c45ab

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What do these values form?

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Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

How does the idea change?

cdbad1a1adaa21bd00dcf9fb0611f095e7bc7e2b

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What's part of the link?

b5ee4301bc285e0b8b4a41fc30ca55d78d3bb5a9

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What are these groups part of?

807f8e5356881fb047aed4d52a205a2b27c04de2

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What is the theory's more improved form?

d17cea12215a4801766a46e271a97032439c1ed5

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What do these values make up?

6c50f8d2d681d27eeb50552d7e0d73e3601ea05d

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What is the theory applied to?

3fa27a6fd3c7b8c679c34361b985f205d5f5f2fe

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What do the Galois groups of certain types of field extensions associated with?

44f040e134999bf916b82613f81e149e62ca918c

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What are matrix groups for any constant c?

f95fbd4500d589ec623ffdfbfc61dca19edea9cf

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What are these groups?

6008bbb3fd75dfa7ba56238125c271e3eb972266

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Where do these groups land?

badac883b1e854a6cd7f71852bf8a45896fe963d

Group_(mathematics)

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

What is the Krull topology?

cb19da658d5a20af768cc47a5ed09d3586befddf

Group_(mathematics)

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Who did Jordan work with?

1bd1cbaf44cd6842ef324c455027a944647317e2

Group_(mathematics)

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

What is Jordan occupation?

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Group_(mathematics)

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

What is not p2?

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Group_(mathematics)

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

What did Borcherds study?

e879bbfe9c84c7cc97ae120631e2d5be3b12d54e

Group_(mathematics)

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

What is the main point of the passage?

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Group_(mathematics)

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

What is an important concept that is mentioned that is important to understand the passage?

f54d42a89bae23b12181d53fa7b5be24bbf38393

Group_(mathematics)

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

What prevents simplification?

c240d0b531e8a45f36b9b0d090c46d5f2cb77846

Group_(mathematics)

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

What did Lagrange study?

ae7b92845fff6061363fd3418d0ddeedc8496d04

Group_(mathematics)

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

What is the main conclusion drawn by the passage?

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Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

What is a part of the new state?

0cd7716eab1c65d11ea5f73c0676c394a914552a

Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

What type of change happens at the Curie temperature when the proper name for the change from a paraelectric to a ferroelectric state occurs

690e3edf0b19fe48c5a719d87b5f29bddc7ce491

Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

What is the proper name for the phase transition point at which 1060 degrees F is reached

a5a5bbcea9d54f20b6101c56f3c25cf6e4f5675d

Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

What is the proper name for the phase transition point at which 570 degrees C is reached

b23099a4ac2a2706f4ca548c9cc5a13404ef326f

Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

Which of the following is not a potential state of ferroelectric materials: paraelectric, cubic or ferroelectric?

97175e07868d9fa5d6a65aef03ea183c7409635a

Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

What happen to the shape in the phase transition besides the change in crystalline form.

c5ba1d583fd1878481e0c982761facb8d99fa461

Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

What happens to the material in the theory?

d1dce0f515b9d452064f38eccdb9bd6e0aa934d5

Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

When does the change occur?

44dd721e4d385ed78a84f8f564571249c31d1ef4

Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

Which of the following is not a potential crystalline form: cubic, paraelectric or tetrahedral?

6f3c00ca71ea88bc5a8cf2f0b15f585534af7d64

Group_(mathematics)

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

Which physical property has a transition point named after a famous female scientist?

10eabce335706b6121871cc4b95c65342d431206

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

Whats needed to understand a group by it's presentation

1e5c4d4e633f1ffc69c52f5507657c2f322cce79

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

What can describe a presentation of every group

8aa0aed7ac1eb0b7a762064f7184455c47b168f7

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

What are quotient and subgroups

6c3409833ff4d29cba3250b1597dd504060e1c95

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

what three words begin the passage?

c8c3bc3423f95937b015fba8923aa210867b0e95

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

what sentence end the passage?

97397795742b693f993570ee0f637632d56a596e

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

The example uses a right and vertical what

58f28dc2b56dab08440f7cecccfa803ef35e3cf5

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

What is the free group

8ab22afea28e3a585c853c9b066774b65ca8fa86

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

what is been discussed?

ad7739a375e6da3819ed4c0c43c78c194568c306

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

what was given as a sample?

9a33cae1dd898912fa16caccded7fe72fd15660a

Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

If the square is symmetrical what does it mean

28f4833d186d20271294e8cefe0e3837dfc110c4

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

To whom was the right to vote extended in 1867?

89c4ae6a92ec5a1e50be008c28afe29e82c47be9

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Between the last time that Victoria attended the State Opening of Parliament prior to 1866 and her attendance at the State Opening of Pariliament in 1866, who passed away?

0a06c2b59b5eb8cb990ee772946b92ed6145432a

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Who was treated like a public meeting, but not like a woman?

13b5bd267f8fe6ade30ef0e600242910618b205c

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Russell followed?

3f2914dd5b73878956548413dfd70aa6f8f65aff

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Who supported extending the franchise to many urban working men, but not to women?

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Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Derby followed?

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Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

What happened in 1868?

7cba1dd2a417f016727d85772029756a83291c4d

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

What happened after Albert's death?

fd7f8f724fca0d0d3088c6851f54227e7d6bd465

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

What year did the person who supported electorates doubling for working men go to Parliment?

b3c19b501914f4cb9cbf1bbd73ac006b06b1fdc1

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Who did not favour votes for women?

08dd059d74257613f0a337e065f2a7c2f5b53deb

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Victoria's gender was?

37f68d5f1c16d96179066cf8d832cfc3faade043

Queen_Victoria

Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Whose rival became prime minister at the end of 1868?

038f4ae08e232bf74e956ce97308d3175010550f

Queen_Victoria

Victoria visited mainland Europe regularly for holidays. In 1889, during a stay in Biarritz, she became the first reigning monarch from Britain to set foot in Spain when she crossed the border for a brief visit. By April 1900, the Boer War was so unpopular in mainland Europe that her annual trip to France seemed inadvisable. Instead, the Queen went to Ireland for the first time since 1861, in part to acknowledge the contribution of Irish regiments to the South African war. In July, her second son Alfred ("Affie") died; "Oh, God! My poor darling Affie gone too", she wrote in her journal. "It is a horrible year, nothing but sadness & horrors of one kind & another."

At what specific point did she do something no one in her position had done before?

47f19052a51f33d33276ff62a4dd4bbe117a99ca

Queen_Victoria

Victoria visited mainland Europe regularly for holidays. In 1889, during a stay in Biarritz, she became the first reigning monarch from Britain to set foot in Spain when she crossed the border for a brief visit. By April 1900, the Boer War was so unpopular in mainland Europe that her annual trip to France seemed inadvisable. Instead, the Queen went to Ireland for the first time since 1861, in part to acknowledge the contribution of Irish regiments to the South African war. In July, her second son Alfred ("Affie") died; "Oh, God! My poor darling Affie gone too", she wrote in her journal. "It is a horrible year, nothing but sadness & horrors of one kind & another."

Where did she end up during special occasions?

a4d138710be83059f788b6d5fb6e4ba0e31b69af

Queen_Victoria

Victoria visited mainland Europe regularly for holidays. In 1889, during a stay in Biarritz, she became the first reigning monarch from Britain to set foot in Spain when she crossed the border for a brief visit. By April 1900, the Boer War was so unpopular in mainland Europe that her annual trip to France seemed inadvisable. Instead, the Queen went to Ireland for the first time since 1861, in part to acknowledge the contribution of Irish regiments to the South African war. In July, her second son Alfred ("Affie") died; "Oh, God! My poor darling Affie gone too", she wrote in her journal. "It is a horrible year, nothing but sadness & horrors of one kind & another."

While where did she do something no one in her position had done before?

311dee289345ebc77a305ffa92c2fc6638587146

Queen_Victoria

Victoria visited mainland Europe regularly for holidays. In 1889, during a stay in Biarritz, she became the first reigning monarch from Britain to set foot in Spain when she crossed the border for a brief visit. By April 1900, the Boer War was so unpopular in mainland Europe that her annual trip to France seemed inadvisable. Instead, the Queen went to Ireland for the first time since 1861, in part to acknowledge the contribution of Irish regiments to the South African war. In July, her second son Alfred ("Affie") died; "Oh, God! My poor darling Affie gone too", she wrote in her journal. "It is a horrible year, nothing but sadness & horrors of one kind & another."

When she had to second guess her usual holiday visits, what was the response?

251545a0baf878164a26f4273a34b30b37b6a224

Queen_Victoria

Victoria visited mainland Europe regularly for holidays. In 1889, during a stay in Biarritz, she became the first reigning monarch from Britain to set foot in Spain when she crossed the border for a brief visit. By April 1900, the Boer War was so unpopular in mainland Europe that her annual trip to France seemed inadvisable. Instead, the Queen went to Ireland for the first time since 1861, in part to acknowledge the contribution of Irish regiments to the South African war. In July, her second son Alfred ("Affie") died; "Oh, God! My poor darling Affie gone too", she wrote in her journal. "It is a horrible year, nothing but sadness & horrors of one kind & another."

What was the thoughts about a certain event made her visits less palatable?

2a765506e798bffcd3930d9d83fec109f1d900c9

Queen_Victoria

In 1887, the British Empire celebrated Victoria's Golden Jubilee. Victoria marked the fiftieth anniversary of her accession on 20 June with a banquet to which 50 kings and princes were invited. The following day, she participated in a procession and attended a thanksgiving service in Westminster Abbey. By this time, Victoria was once again extremely popular. Two days later on 23 June, she engaged two Indian Muslims as waiters, one of whom was Abdul Karim. He was soon promoted to "Munshi": teaching her Hindustani, and acting as a clerk. Her family and retainers were appalled, and accused Abdul Karim of spying for the Muslim Patriotic League, and biasing the Queen against the Hindus. Equerry Frederick Ponsonby (the son of Sir Henry) discovered that the Munshi had lied about his parentage, and reported to Lord Elgin, Viceroy of India, "the Munshi occupies very much the same position as John Brown used to do." Victoria dismissed their complaints as racial prejudice. Abdul Karim remained in her service until he returned to India with a pension on her death.

Who was less highly ranked, Frederick Ponsonby or Lord Elgin?

8028eb1bb592938fabad5a8301be87929105668c

Queen_Victoria

In 1887, the British Empire celebrated Victoria's Golden Jubilee. Victoria marked the fiftieth anniversary of her accession on 20 June with a banquet to which 50 kings and princes were invited. The following day, she participated in a procession and attended a thanksgiving service in Westminster Abbey. By this time, Victoria was once again extremely popular. Two days later on 23 June, she engaged two Indian Muslims as waiters, one of whom was Abdul Karim. He was soon promoted to "Munshi": teaching her Hindustani, and acting as a clerk. Her family and retainers were appalled, and accused Abdul Karim of spying for the Muslim Patriotic League, and biasing the Queen against the Hindus. Equerry Frederick Ponsonby (the son of Sir Henry) discovered that the Munshi had lied about his parentage, and reported to Lord Elgin, Viceroy of India, "the Munshi occupies very much the same position as John Brown used to do." Victoria dismissed their complaints as racial prejudice. Abdul Karim remained in her service until he returned to India with a pension on her death.

who participated in a procession?

2e495163d0ba36abb0d67511fbeb56de0ebd3f1c

Queen_Victoria

In 1887, the British Empire celebrated Victoria's Golden Jubilee. Victoria marked the fiftieth anniversary of her accession on 20 June with a banquet to which 50 kings and princes were invited. The following day, she participated in a procession and attended a thanksgiving service in Westminster Abbey. By this time, Victoria was once again extremely popular. Two days later on 23 June, she engaged two Indian Muslims as waiters, one of whom was Abdul Karim. He was soon promoted to "Munshi": teaching her Hindustani, and acting as a clerk. Her family and retainers were appalled, and accused Abdul Karim of spying for the Muslim Patriotic League, and biasing the Queen against the Hindus. Equerry Frederick Ponsonby (the son of Sir Henry) discovered that the Munshi had lied about his parentage, and reported to Lord Elgin, Viceroy of India, "the Munshi occupies very much the same position as John Brown used to do." Victoria dismissed their complaints as racial prejudice. Abdul Karim remained in her service until he returned to India with a pension on her death.

who acted as a clerk?

ecdc903a8d4031e384d8dcbd8738873f22f72e2e

Queen_Victoria

In 1887, the British Empire celebrated Victoria's Golden Jubilee. Victoria marked the fiftieth anniversary of her accession on 20 June with a banquet to which 50 kings and princes were invited. The following day, she participated in a procession and attended a thanksgiving service in Westminster Abbey. By this time, Victoria was once again extremely popular. Two days later on 23 June, she engaged two Indian Muslims as waiters, one of whom was Abdul Karim. He was soon promoted to "Munshi": teaching her Hindustani, and acting as a clerk. Her family and retainers were appalled, and accused Abdul Karim of spying for the Muslim Patriotic League, and biasing the Queen against the Hindus. Equerry Frederick Ponsonby (the son of Sir Henry) discovered that the Munshi had lied about his parentage, and reported to Lord Elgin, Viceroy of India, "the Munshi occupies very much the same position as John Brown used to do." Victoria dismissed their complaints as racial prejudice. Abdul Karim remained in her service until he returned to India with a pension on her death.

What holiday did Victoria celebrate the day after her fiftieth anniversary banquet?

9576b9980a4faf2a952417835f8701b5e2447edb

Queen_Victoria

In 1887, the British Empire celebrated Victoria's Golden Jubilee. Victoria marked the fiftieth anniversary of her accession on 20 June with a banquet to which 50 kings and princes were invited. The following day, she participated in a procession and attended a thanksgiving service in Westminster Abbey. By this time, Victoria was once again extremely popular. Two days later on 23 June, she engaged two Indian Muslims as waiters, one of whom was Abdul Karim. He was soon promoted to "Munshi": teaching her Hindustani, and acting as a clerk. Her family and retainers were appalled, and accused Abdul Karim of spying for the Muslim Patriotic League, and biasing the Queen against the Hindus. Equerry Frederick Ponsonby (the son of Sir Henry) discovered that the Munshi had lied about his parentage, and reported to Lord Elgin, Viceroy of India, "the Munshi occupies very much the same position as John Brown used to do." Victoria dismissed their complaints as racial prejudice. Abdul Karim remained in her service until he returned to India with a pension on her death.

What title did Victoria hold?

0c7bfaec0462090dd7288ed675935008375f4fe2

Queen_Victoria

In 1887, the British Empire celebrated Victoria's Golden Jubilee. Victoria marked the fiftieth anniversary of her accession on 20 June with a banquet to which 50 kings and princes were invited. The following day, she participated in a procession and attended a thanksgiving service in Westminster Abbey. By this time, Victoria was once again extremely popular. Two days later on 23 June, she engaged two Indian Muslims as waiters, one of whom was Abdul Karim. He was soon promoted to "Munshi": teaching her Hindustani, and acting as a clerk. Her family and retainers were appalled, and accused Abdul Karim of spying for the Muslim Patriotic League, and biasing the Queen against the Hindus. Equerry Frederick Ponsonby (the son of Sir Henry) discovered that the Munshi had lied about his parentage, and reported to Lord Elgin, Viceroy of India, "the Munshi occupies very much the same position as John Brown used to do." Victoria dismissed their complaints as racial prejudice. Abdul Karim remained in her service until he returned to India with a pension on her death.

Who was less highly ranked, Lord Elgin or Queen Victoria?

b6507a3d75e97130a69e6642400835dc16bfdf13

Queen_Victoria

In 1887, the British Empire celebrated Victoria's Golden Jubilee. Victoria marked the fiftieth anniversary of her accession on 20 June with a banquet to which 50 kings and princes were invited. The following day, she participated in a procession and attended a thanksgiving service in Westminster Abbey. By this time, Victoria was once again extremely popular. Two days later on 23 June, she engaged two Indian Muslims as waiters, one of whom was Abdul Karim. He was soon promoted to "Munshi": teaching her Hindustani, and acting as a clerk. Her family and retainers were appalled, and accused Abdul Karim of spying for the Muslim Patriotic League, and biasing the Queen against the Hindus. Equerry Frederick Ponsonby (the son of Sir Henry) discovered that the Munshi had lied about his parentage, and reported to Lord Elgin, Viceroy of India, "the Munshi occupies very much the same position as John Brown used to do." Victoria dismissed their complaints as racial prejudice. Abdul Karim remained in her service until he returned to India with a pension on her death.

What did Abdul Karim deceive Victoria regarding?

710142901512b3ba4225285581519c9b797440e5

Queen_Victoria

In 1887, the British Empire celebrated Victoria's Golden Jubilee. Victoria marked the fiftieth anniversary of her accession on 20 June with a banquet to which 50 kings and princes were invited. The following day, she participated in a procession and attended a thanksgiving service in Westminster Abbey. By this time, Victoria was once again extremely popular. Two days later on 23 June, she engaged two Indian Muslims as waiters, one of whom was Abdul Karim. He was soon promoted to "Munshi": teaching her Hindustani, and acting as a clerk. Her family and retainers were appalled, and accused Abdul Karim of spying for the Muslim Patriotic League, and biasing the Queen against the Hindus. Equerry Frederick Ponsonby (the son of Sir Henry) discovered that the Munshi had lied about his parentage, and reported to Lord Elgin, Viceroy of India, "the Munshi occupies very much the same position as John Brown used to do." Victoria dismissed their complaints as racial prejudice. Abdul Karim remained in her service until he returned to India with a pension on her death.

who taught Victoria Hindustani?