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Math Help - direct products problem

  1. #1
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    direct products problem

    Suppose that G is a finite group such that for each subgroup H of G, there is a homomorphism T:G \to Hsuch that [MATHT(h)=h[\MATH]for all h \in H. Show that G is a product of groups of prime order.
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    Quote Originally Posted by Chandru1 View Post
    Suppose that G is a finite group such that for each subgroup H of G, there is a homomorphism T:G \to Hsuch that [MATHT(h)=h[\MATH]for all h \in H. Show that G is a product of groups of prime order.

    Be sure you understand and can prove the following (check Rotman's book under "minimal normal subgroups", for example):

    == Every finite groups has non-trivial minimal normal subgroups (mns's)

    == Every mns is either simple or a direct product of mutually isomorphic simple groups

    == if N is a normal sbgp. of G and f: G --> N is a retract (this is what you defined in your question), then G = NK, where K = ker f, and this in fact is a direct product since K is a normal complement of N)

    So choose some non-trivial mns N of G (if N = G then G is simple but then it is either abelian of prime order and we're done or else it is non-abelian simple, which is impossible since then the condition isn't fulfilled since G then has no non-trivial homomorphism because it has no non-trivial normals sbgps.!).
    From the above we can write G = N x K. Now take a mns N1 of K, then N1 is normal in G (can you see why?) and again we have a normal
    complement K1 of N1 in G, so G = N1 x K1 but since N 1 < K we in fact get K = N1 x (K /\ K1) ==> G = N x N1 x (K /\ K1).
    Continue as above now with K2 = K /\ K1 and etc.
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