Cauchy's Theorem Problem

1. Nov 26, 2007

J0EBL0W

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. Nov 27, 2007

HallsofIvy

Staff Emeritus
What is "f"? Did you mean A(P)?

3. Nov 27, 2007

Dick

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.

4. Nov 27, 2007

J0EBL0W

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.

5. Nov 27, 2007

ZioX

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