Math 411 Homework Assignment 7 Due 3/24/2017 by the end of lecture 1. (a) Show that the cycle (a1 a2 . . . ak ) Sn is even if and only if k is odd. (b) Suppose = (a1 a2 ) is a transposition and let Sn . Show that 1 is a transposition. 2. Let G be a subgroup of Sn that contains an odd permutation . (a) Prove that the number of even permutations in G is the same as the number of odd permutations in G. (Hint: Imitate the proof that |An | = |S2n | .) (b) Use part (a) to conclude that |G| is divisible by 2. 3. Suppose n 3 is an integer and consider the dihedral group Dn = {e, r, . . . , rn1 , s, rs, . . . , rn1 s}, where r is the rotation of order n, s is the reflection of order 2, and r and s satisfy the relation srs1 = r1 . The goal of this problem is to show that the center Z(Dn ) of Dn has the following description: Z(Dn ) = {e} if n is odd, and Z(Dn ) = {e, rn/2 } if n is even. (a) Prove by induction that srk = rk s for any integer k. (b) Show that g Z(Dn ) if and only if g commutes with r and s, i.e. gr = rg and gs = sg. (c) Suppose that g = ri sj Z(Dn ). Note that j = 0 or j = 1 since s has order 2. Use the condition gr = rg to conclude that j = 0. (Hint: Show that if j = 1, then sr = rs. Use the fact that n 3 together with the relation sr = r1 s to obtain a contradiction.) Thus, g = ri . (d) Use the condition gs = sg to conclude that n divides 2i. (Hint: Recall that r has order n.) (e) Show that i = 0 or 2i = n. Conclude that g = e if n is odd and g = e or g = rn/2 if n is even. (Hint: Notice that by construction, we have 0 i n 1.) 4. Show that the symmetric group S4 is not isomorphic to the dihedral group D12 . (You may use without proof the fact that Sn has trivial center for n 3.) 5. Show that the quaternion group Q8 is not isomorphic to the dihedral group D4 . (Hint: How many elements of order 4 does each group have?) 6. Suppose p is a prime and G is a cyclic group of order p3 . Show that G cannot be written as a nontrivial direct product G = H K, with H and K subgroups of G and |H|, |K| > 1. (Hint: If G = H K, what is the largest possible order of an element of G?)