1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted