Symmetry Groups

    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 symmetry groups _gr_1.gif] denote the distance between any two points in the plane symmetry groups _gr_2.gif]. Any permutation symmetry groups _gr_3.gif] of the plane is said to preserve distance when

symmetry groups _gr_4.gif]

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 symmetry groups _gr_5.gif]

Proposition (Isometries) The set symmetry groups _gr_6.gif] is a subgroup of symmetry groups _gr_7.gif] with composition as the operation.

    Proof. Since the identity mapping preserves distance symmetry groups _gr_8.gif] is nonempty. Let symmetry groups _gr_9.gif] It suffices to show that symmetry groups _gr_10.gif] or equivalently,

symmetry groups _gr_11.gif]

Since symmetry groups _gr_12.gif] it follows that symmetry groups _gr_13.gif] symmetry groups _gr_14.gif] symmetry groups _gr_15.gif] symmetry groups _gr_16.gif] symmetry groups _gr_17.gif] as desired. symmetry groups _gr_18.gif]

Definition (Invariance) Let symmetry groups _gr_19.gif] be any nonempty subset and symmetry groups _gr_20.gif] with symmetry groups _gr_21.gif] Then

symmetry groups _gr_22.gif]

and we say that elements of symmetry groups _gr_23.gif] leave symmetry groups _gr_24.gif] elementwise invariant. Further,

symmetry groups _gr_25.gif]

and we say that elements of symmetry groups _gr_26.gif] are symmetry groups _gr_27.gif]-invariant.

Proposition (Invariant Subgroups) Let symmetry groups _gr_28.gif] be any nonempty set and symmetry groups _gr_29.gif] with symmetry groups _gr_30.gif] Then symmetry groups _gr_31.gif] and symmetry groups _gr_32.gif] are subgroups of symmetry groups _gr_33.gif] and symmetry groups _gr_34.gif] is a subgroup of symmetry groups _gr_35.gif]

    Proof.  The identity mapping is in both symmetry groups _gr_36.gif] and symmetry groups _gr_37.gif] To show that symmetry groups _gr_38.gif] is a subgroup of symmetry groups _gr_39.gif], let symmetry groups _gr_40.gif] and symmetry groups _gr_41.gif] Then symmetry groups _gr_42.gif] symmetry groups _gr_43.gif] symmetry groups _gr_44.gif] symmetry groups _gr_45.gif] and so, symmetry groups _gr_46.gif] is a subgroup of symmetry groups _gr_47.gif]   To show that symmetry groups _gr_48.gif] is a subgroup of symmetry groups _gr_49.gif], let symmetry groups _gr_50.gif] Then symmetry groups _gr_51.gif] symmetry groups _gr_52.gif] symmetry groups _gr_53.gif] symmetry groups _gr_54.gif] and so, symmetry groups _gr_55.gif] is a subgroup of symmetry groups _gr_56.gif]  If symmetry groups _gr_57.gif] then symmetry groups _gr_58.gif] for all symmetry groups _gr_59.gif] and so symmetry groups _gr_60.gif] Therefore, symmetry groups _gr_61.gif] and so symmetry groups _gr_62.gif] is a subgroup of   symmetry groups _gr_63.gif] as desired. symmetry groups _gr_64.gif]

Definition (Symmetry Group) If symmetry groups _gr_65.gif] is a set of points in a plane symmetry groups _gr_66.gif], then symmetry groups _gr_67.gif] the group of all motions of the plane leaving symmetry groups _gr_68.gif] invariant, is called the symmetry group of symmetry groups _gr_69.gif]

Example (Symmetry Group of the Parallelogram) The only rigid motions of the parallelogram are the identity and a symmetry groups _gr_70.gif] clockwise rotation about symmetry groups _gr_71.gif].  

symmetry groups _gr_72.gif]
Let symmetry groups _gr_73.gif] be the identity and symmetry groups _gr_74.gif] be the symmetry groups _gr_75.gif] clockwise rotation around symmetry groups _gr_76.gif] Then the Cayley table for the symmetry group of the parallelogram  is  

symmetry groups _gr_77.gif]
symmetry groups _gr_78.gif]

Example (Symmetry Group of the Rectangle) The only rigid motions of the rectangle are the identity, symmetry groups _gr_79.gif] clockwise  rotation about symmetry groups _gr_80.gif] reflection through symmetry groups _gr_81.gif] and symmetry groups _gr_82.gif]  

symmetry groups _gr_83.gif]
Let symmetry groups _gr_84.gif] be the identity, symmetry groups _gr_85.gif] be the symmetry groups _gr_86.gif] clockwise rotation around symmetry groups _gr_87.gif] and symmetry groups _gr_88.gif] and symmetry groups _gr_89.gif] be the relections through symmetry groups _gr_90.gif] and symmetry groups _gr_91.gif], respectively. Then the Cayley table for the symmetry group of the rectangle  is

symmetry groups _gr_92.gif]  
symmetry groups _gr_93.gif]

Example (Symmetry Group of the Square) It suffices to consider only rotations and reflections. Let symmetry groups _gr_94.gif] represent a clockwise rotation around symmetry groups _gr_95.gif] of symmetry groups _gr_96.gif]0, 90, 180, 270 degrees, symmetry groups _gr_97.gif] and symmetry groups _gr_98.gif] be the reflections through the lines symmetry groups _gr_99.gif] and symmetry groups _gr_100.gif] respectively, and let symmetry groups _gr_101.gif] and symmetry groups _gr_102.gif] be the relections through the diagonals symmetry groups _gr_103.gif] and symmetry groups _gr_104.gif] respectively.   

symmetry groups _gr_105.gif]

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

symmetry groups _gr_106.gif]
symmetry groups _gr_107.gif]

Example (Symmetry Group of an Equilateral Triangle) The rigid motions of an equilateral traingle yield the group symmetry groups _gr_108.gif] The clockwise rotations are given by the permutations symmetry groups _gr_109.gif] symmetry groups _gr_110.gif] and symmetry groups _gr_111.gif] Reflecting the triangle about one of the angle bisectors gives one of the permutations symmetry groups _gr_112.gif] symmetry groups _gr_113.gif] or symmetry groups _gr_114.gif]

symmetry groups _gr_115.gif]

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

symmetry groups _gr_116.gif]
symmetry groups _gr_117.gif]

Cite this as:
Symmetry Groups
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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