MATH Algebra for High School Teachers Dihedral & Symmetric Groups

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1 MATH Algebra for High School Teachers Dihedral & Symmetric Groups Professor Donald L White Department of Mathematical Sciences Kent State University DL White (Kent State University 1 / 11

2 Dihedral Groups The group of symmetries of the square is the dihedral group of order 8, denoted D 4 Recall that D 4 contains 4 rotations and 4 reflections More generally, a regular n-gon has n rotations through multiples of 360 n degrees, and n reflections across its n axes of symmetry Definition The group of symmetries of the regular n-gon (for n 3 is called the dihedral group of order 2n and is denoted D n NOTE: Groups are sometimes named for their order (number of elements and sometimes for the size of the object they act on Accordingly, some sources use D n to denote the dihedral group of order 2n, (as in our text while others use D 2n Thus, depending on the source, the group of symmetries of the square may be called D 4 or D 8 DL White (Kent State University 2 / 11

3 Symmetries of the Pentagon Example 1: Symmetries of the regular pentagon The central angle of a regular pentagon is degrees; that is, 72 : Therefore, the dihedral group D 5 contains 5 rotations, through 0, 72, 144, 216, and 288 For example, the 144 rotation corresponds to the permutation of vertices ( DL White (Kent State University 3 / 11

4 Symmetries of the Pentagon [Example 1, continued] The pentagon has no pairs of opposite sides or pairs of opposite vertices In this case, each axis of symmetry is a line through a vertex and the center of the opposite side: For example, the reflection across the axis through vertex 2 is or ( DL White (Kent State University 4 / 11

5 Symmetries of the Hexagon Example 2: Symmetries of the regular hexagon The central angle of a regular hexagon is degrees; that is, 60 : Therefore, the dihedral group D 6 contains 6 rotations, through 0, 60, 120, 180, 240, and 300 ; eg, the 120 rotation corresponds to the permutation of vertices ( DL White (Kent State University 5 / 11

6 Symmetries of the Hexagon [Example 2, continued] The hexagon has 3 pairs of opposite vertices and 3 pairs of opposite sides Each axis of symmetry is a line through one of these pairs DL White (Kent State University 6 / 11

7 Symmetries of the Hexagon [Example 2, continued] For example, the reflection across the axis through vertices 2 and 5 is or ( The reflection across the horizontal axis of symmetry is or ( DL White (Kent State University 7 / 11

8 Symmetric Groups For n 4, not every permutation of the vertices of an n-gon is in D n For example, if n = 5, then the permutation ( of the vertices of is not a rigid motion, hence is not in D 5 Therefore, for n 4, D n is a proper subset of the set of all permutations of {1, 2, 3,, n} (ie, the set of one-to-one, onto functions from {1, 2, 3,, n} to itself Definition The set of all permutations of the set {1, 2, 3,, n} is the symmetric group on n letters, denoted S n DL White (Kent State University 8 / 11

9 Symmetric Groups As before, we can write the permutations in S n in two-row form: ( n, a 1 a 2 a 3 a 4 a n where a 1, a 2,, a n are the integers 1, 2,, n in some order Since all one-to-one, onto functions are permitted, there are n choices for a 1 n 1 choices for a 2 n 2 choices for a 3 2 choices for a n 1 1 choice for a n All choices are independent, so the total number of choices is n(n 1(n 2 (2(1 = n!, hence S n = n! As D n = 2n, we have D n S n for n 4 Note, however, that 3! = 2 3, so in fact D 3 = S 3 DL White (Kent State University 9 / 11

10 Symmetric Groups We can again combine two permutations by function composition For example, if ( f = and g = in S 7, then (composing right to left as usual f g = ( ( ( = ( Composition of functions is always associative Therefore, this binary operation on S n is associative DL White (Kent State University 10 / 11

11 Symmetric Groups The permutation i = ( n n is an identity element for S n ; that is, f i = i f = f for all f in S n Every one-to-one, onto function f has an inverse function f 1 ; that is, f f 1 = f 1 f = i Therefore, every element of S n has an inverse in S n The inverse of f S n is easy to find: turn the two-row form upside down For example, in S 7, if then f 1 = f = ( ( =, ( DL White (Kent State University 11 / 11