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Cauchy's Theorem Problem

  1. Nov 26, 2007 #1
    Cauchy's Theorem Problem (Abstract Algebra question)

    1. The problem statement, all variables and given/known data
    I've been thinking about this problem for a couple days now, and I don't even know how to approach it. The problem is:
    Let G be a group of order (p^n)*m, where p is a prime and p does not divide m. Suppose that G has a normal subgroup P of order p^n. Prove that f(P)=P for every automorphism 'A' of G.

    I can't even convince myself that the question is true, then alone a method on how to show it. Any point in the right direction would help me a ton. Thanks.


    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Nov 26, 2007
  2. jcsd
  3. Nov 27, 2007 #2

    HallsofIvy

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    What is "f"? Did you mean A(P)?
     
  4. Nov 27, 2007 #3

    Dick

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    Prove that for g in G, the order of g divides p^n iff g is an element of P. One direction is easy, for the other direction take g not in P. Then gP is an element of the quotient group G/P which has order n. Can you finish? Once you have that, automorphisms preserve the order of elements.
     
  5. Nov 27, 2007 #4
    I have no problem showing the orders of G and G/P, but I don't understand why an automorphism automatically preserve the order of elements.
     
  6. Nov 27, 2007 #5
    Automorphisms are isomorphisms. Do you believe that isomorphisms preserve the order of elements? You should.

    Edit: Hopefully you realize that when I said that 'automorphisms are isomorphisms' I was not implying that they are the same thing. Automorphisms are isomorphisms that map the group back onto the same group. The order of the automorphism group gives a sense of symmetry that the group has.
     
    Last edited: Nov 27, 2007
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