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Another Baby Herstein algebra problem

  1. Jan 22, 2008 #1
    1. The problem statement, all variables and given/known data
    Let G be a group, |G|=m*p^n, p prime, H a normal subgroup of G, |H| = p^n, gcd(p,m) = 1. Show that H is a characteristic subgroup of G. (Definition of characteristic: any automorphism of G fixes H, i.e. when restricted to H it is an automorphism of H.)

    2. Relevant equations

    3. The attempt at a solution
    This is one of those situations similar to the last post. It's obviously true with some later results in hand, but the problem is given prior to the presentation of those theorems.

    The outline of my "proof": Show that any element of order p^k is in H. Proceed by assuming you have a g in G that is of order p^k, and g is not in H. Then construct a subgroup of G which has order p^r, where r > n. Since this is impossible g is in H, and H is characteristic.

    This almost works with what I have, save for one problem. My strategy relies on the fact that my subgroup < g > H has an order a power of p greater than what is possible. Yet to prove that I need to use the fact that |< g >H| = |< g >|*|H|/|< g > intersect H|. That result is not available until later. Can anyone suggest an alternative, more direct strategy? Thanks.
  2. jcsd
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