Question

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, any two Sylow p-
subgroups of G are conjugate in G.

3. np ≡ 1 (mod p) and np|m.

Answer 1