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Math Help - normal p-subgroups of a finite group

  1. #1
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    normal p-subgroups of a finite group

    Let G be a finite group. Let H and K be normal p-subgroups of G for some prime p with H being a proper subgroup of K and H/K is elementary abelian p-group. Let S be a normal subgroup of K with H<S. If H contain every element of order p of S and  K=<S^{g},g \in G>, then H contain every element of order p of K.

    I need to prove the above statement. Thanks in advance.
    Last edited by moont14263; August 17th 2012 at 10:41 AM.
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  2. #2
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    Re: normal p-subgroups of a finite group

    your question doesn't make sense, as written. if H < K, then what does H/K mean?
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  3. #3
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    Re: normal p-subgroups of a finite group

    I fix the mistake in the question with some additions.
    Let G be a finite solvable group. Let K/H be a chief factor of G that is not of prime order, where K is a p-subgroup of G for some prime p divides the order of G. Let S be a proper normal subgroup of K with H <S and |S/H|=p. If H contain every element of order p of S and K=<S^{g},g \in G>, then H contains every element of order p of K.

    I need to prove the above statement.

    Here is what I know.

    Since G is solvable then K/H is abelian p-group of exponent p. \bigcap_{g \in G}S^{g} is a normal subgroup of G that contains H. So H=  \bigcap_{g \in G}S^{g}.

    Thanks in advance.
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