Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on. COUNT 5
QSN1 What are groups consisting of symmetries of given arithmetic concepts? ANS1 Symmetry groups QSN2 What term describes the introductory symmetry group of the square? ANS2 geometric nature QSN3 What can regarded as the study of symmetry? ANS3 group theory QSN4 What is the name of the rule that must be met for a group operation to occur? ANS4 group law QSN5 The group pattern is connected to the structure of the target by what behavior? ANS5 group action