Normal closure (group theory)

In group theory, the normal closure of a subset "S" of a group "G" is the smallest normal subgroup of "G" containing "S".
Properties and description.
Formally, if "G" is a group and "S" is a subset of "G", the normal closure of formula_1 of "S" is the intersection of all normal subgroups of "G" containing "S":
formula_2
The normal closure formula_1 is the smallest normal subgroup of "G" containing "S", in the sense that formula_1 is a subset of every normal subgroup of "G" that contains "S".
The subgroup formula_1 is generated by the set formula_6 of all conjugates of elements of "S" in "G".
Therefore one can also write
formula_7
Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set formula_8 is the trivial subgroup.
A variety of other notations are used for the normal closure in the literature, including formula_9, formula_10, formula_11, and formula_12.
Dual to the concept of normal closure is that of "normal interior" or "normal core", defined as the join of all normal subgroups contained in "S".
Group presentations.
For a group "G" given by a presentation formula_13 with generators "S" and defining relators "R", the presentation notation means that "G" is the quotient group formula_14, where formula_15 is a free group on "S".