26 Chapter 1. Groups I Example 1.31. The symmetry group Σ(π5) of a regular pentagon π5 with vertices v1,... , v5 and center O (Figure 1.6) has 10 elements: the rotations about the origin by (72j)◦, where 0 ≤ j ≤ 4, as well as the reflections in the lines Ovk for 1 ≤ k ≤ 5. The symmetry group Σ(π5) is called the dihedral group of order 10, and it is denoted by D10. m v1 v3 v2 v4 m 1 2 O v5 m 3 m 4 m 5 Figure 1.6. Pentagon. m v1 v v2 v m 1 2 3 4 O v5 m 3 v6 Figure 1.7. Hexagon. Definition. If πn is a regular polygon with n ≥ 3 vertices v1,v2,...,vn and center O, then the symmetry group Σ(πn) is called the dihedral group of order 2n, and it is denoted16 by D2n. We define the dihedral group D4 = V, the four-group, to be the group of order 4 V = (1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) ⊆ S4 [see Example 1.33(i) on page 28]. Remark. Some authors define the dihedral group D2n as a group of order 2n generated by elements a, b such that an = 1, b2 = 1, and bab = a−1. Of course, one is obliged to prove existence of such a group this is done in Proposition 4.86. The dihedral group D2n of order 2n contains the n rotations ρj about the center by (360j/n)◦, where 0 ≤ j ≤ n−1. The description of the other n elements depends on the parity of n. If n is odd (as in the case of the pentagon see Figure 1.6), then the other n symmetries are reflections in the distinct lines Ovi, for i = 1, 2,...,n. If n = 2q is even (the square in Figure 1.5 or the regular hexagon in Figure 1.7), then each line Ovi coincides with the line Ovq+i, giving only q such reflections the remaining q symmetries are reflections in the lines Omi for i = 1, 2,...,q, where mi is the midpoint of the edge vivi+1. For example, the six lines of symmetry of π6 are Ov1, Ov2, and Ov3, and Om1, Om2, and Om3. Exercises 1.27. Let G be a semigroup. Prove directly, without using generalized associativity, that (ab)(cd) = a[(bc)d] in G. 16Some authors denote D2n by Dn.

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