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Homework Help: Fully characteristic subgroups

  1. Dec 28, 2008 #1
    Let G=Z_2XS_3 (Z_2:cyclic group of order 2; S_3: Symmetric group on 3) . Show Center of G, Z(G) is not a fully characteristic (or invariant) subgroup of G.

    Apparently, Z(G)=Z_2
    I know that I need to show that there exists an endomorphism g from G to G such that g(Z_2) is not contained in Z_2.
    But I am not sure how.

    Also, to prove that every fully characteristic subgroup H is also characteristic, I now how to show that for every automorphism p in Aut(G), p(H) is contained in H, but for some reason I don't see why H is contained in p(H).
  2. jcsd
  3. Dec 30, 2008 #2


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    There are subgroups of S_3 that are isomorphic to Z_2. Map the center to one of those. For the second question, p^(-1) is also in Aut(G).
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