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Let G be a group of order p am where p is a prime not dividing...

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Advance Math

Let G be a group of order p

am where p is a prime not dividing m. Show the following

1. Sylow p-subgroups of G exist; i.e. Sylp(G) 6= ?.
2. If P ? Sylp(G) and Q is any p-subgroup of G, then there exists g? G such that Q 6
gP g?1

; i.e. Q is contained in some conjugate of P. In particular, anytwo Sylow p-
subgroups of G are conjugate in G.

3. np ? 1 (mod p) and np|m.

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