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 G is a finite ring and p is a prime number such that p divides the order of G for k times. since p is a prime less than the order of G , there exists an element a in G such that (ap) k = n where n is the order of G. there exists an element (a m ) p such that ( a m )p* (am)q = 0 so, (a m )p and (am)q are the zero divisors in Zp G. ∴ Zp(G) is not an integral domain.