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Abstract Algebra question

  1. May 1, 2010 #1
    1. The problem statement, all variables and given/known data
    (1)To prove this I have to let G be a group, with |G|=p^2.
    (2)Use the G/Z(G) theorem to show G must be Abelian.
    (3) Use the Fundamental Theorem of Finite Abelian Groups to find all the possible isomorphism types for G.

    2. Relevant equations
    Z(G) = the center of G (a is an element of G such that ax=xa for all x in G)

    3. The attempt at a solution
    I can prove it by using Conjugacy classes and gettting that the order of Z(G) must be non-trival and going on from there, but we have not gotten to Conjugacy Classes yet so i can't use this fact. Can anyone help me on this? I know that |Z(G)|=1 or pq when the order of |G|=pq where p and q are not distinct primes. From there I am unsure on how to uise the G/Z(G) thereom to prove that G is abelian.
  2. jcsd
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