C Roettger Spring 2013 Math 301 – Homework 5 Problem 6.32 Suppose that G is a finite Abelian group and that G has no element of order 2. Show that the mapping g 7→ g 2 is an automorphism of G. Show, by example, that if G is infinite the mapping need not be an automorphism. Problem 6.34 If a and g are elements of a group, prove that C(a) is isomorphic to C(gag −1 ). Problem 7.10 Let a and b be non-identity elements of different orders in a group G of order 155. Prove that the only subgroup of G that contains a and b is G itself. Problem 7.36 Let G be a group of order pa where p is prime. Prove that the center of G cannot have order pa−1 .