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hsong9
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Prove that the group ring ZpG is not a domain.
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.
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.
Homework Statement
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.
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.