Prove that the group ring ZpG is not a domain. 1. The problem statement, all variables and given/known data Let G be a finite group and let p >= 3 be a prime such that p | |G|. Prove that the group ring ZpG is not a domain. Hint: Think about the value of (g − 1)p in ZpG where g in G and where 1 = e in G is the identity element of G. 3. The attempt at a solution Suppose that ZpG is a domain. Find some g in G with order p. Note that g is not 1. (g-1)^p = g^p - 1 = 1 - 1 = 0 However, since we assumed that ZpG is a domain, it follows that g-1 = 0, so that g=1 - a contradiction.