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## Homework Statement

Let G be a finite group and let p >= 3 be a prime such that p | |G|.

Prove that the group ring Z

_{p}G 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.

## 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 (a

^{p})

^{k}= n where n is the order of G.

there exists an element (a

^{m })

^{ p}such that ( a

^{m})

^{p}* (a

^{m})

^{q}= 0

so, (a

^{m})

^{p}and (a

^{m})

^{q}are the zero divisors in Z

_{p}G.

∴ Z

_{p}(G) is not an integral domain.