Adventures in Algebra VIII: Homomorphisms 1 Let Dan be the symmetry group of the 2n)-gon. Remember...
proof should involve r^(2i) for some i Adventures in Algebra VII: This is completely normal. 1 Let n be a positive even integer. Recall that the dihedral group D is generated by r ands subject to r" = s? = c and rs = sr-1. Show that (-2) is a normal subgroup of Dm
2. Let n 2 3, and G D2n e,r,r2,... ,r"-1,s, sr, sr2,..., sr-'), the dihedral group with 2n ele- 3, ST, ST,..,ST ments. We let R-(r) denote the subgroup consisting of all rotations. (a) Show that, if M is a subgroup of R, and is in GR, then the union M UrM is a subgroup of G. Here xM-{rm with m in M) (b) Now take n- 12 and M (). How many distinct subgroups does the construction in (a)...