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Prove that the group ring Z p G is not a domain.

  1. May 6, 2009 #1
    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.
     
  2. jcsd
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