Symmetry Group

    Groups of symmetries are very useful in geometry. This topic looks at how to associate a group with each figure in a plane. By defining isometries as permutations of the plane that preserves distance, it is shown that the set of all isometries (or rigid motions) forms a subgroup of the permutation group of the plane. Symmetry groups are then defined and some examples are given such as the symmetry group of the rectangle, square, and equilateral triangle.

Definition (Preserving Distance) Let denote the distance between any two points in the plane . Any permutation of the plane is said to preserve distance when

Example (Preserving Distance) Rotations about a fixed point, the reflection about a fixed line, and translations that send every point in the plane a fixed distance in the same direction are all examples of distance preserving permutations of the plane.

Definition (Isometries) The set of all permutations of the plane that preserve distance between points is called the isometries (or motions) of the plane and is denoted by

Proposition (Isometries) The set is a subgroup of with composition as the operation.

    Proof. Since the identity mapping preserves distance is nonempty. Let It suffices to show that or equivalently,



Since it follows that as desired.

Definition (Invariance) Let be any nonempty subset and with Then



and we say that elements of leave elementwise invariant. Further,



and we say that elements of are -invariant.

Proposition (Invariant Subgroups) Let be any nonempty set and with Then and are subgroups of and is a subgroup of

    Proof.  The identity mapping is in both and To show that is a subgroup of , let and Then and so, is a subgroup of    To show that is a subgroup of , let Then and so, is a subgroup of   If then for all and so Therefore, and so is a subgroup of   as desired.

Definition (Symmetry Group) If is a set of points in a plane , then the group of all motions of the plane leaving invariant, is called the symmetry group of

Example (Symmetry Group of the Parallelogram) The only rigid motions of the parallelogram are the identity and a clockwise rotation about .  


Let be the identity and be the clockwise rotation around Then the Cayley table for the symmetry group of the parallelogram  is  

Example (Symmetry Group of the Rectangle) The only rigid motions of the rectangle are the identity, clockwise  rotation about reflection through and   


Let be the identity, be the clockwise rotation around and and be the relections through and , respectively. Then the Cayley table for the symmetry group of the rectangle  is

  

Example (Symmetry Group of the Square) It suffices to consider only rotations and reflections. Let represent a clockwise rotation around of 0, 90, 180, 270 degrees, and be the reflections through the lines and respectively, and let and be the relections through the diagonals and respectively.   



Then the Cayley table for the symmetry group of the square is  

Example (Symmetry Group of an Equilateral Triangle) The rigid motions of an equilateral traingle yield the group The clockwise rotations are given by the permutations and Reflecting the triangle about one of the angle bisectors gives one of the permutations or



Then the Cayley table for the symmetry group of the equilateral triangle is  

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