1. The problem statement, all variables and given/known data Let G be a finite group, H a normal subgroup of G and B a p-Sylow subgroup of H for some p dividing |H|. 1. Let q be a prime dividing |G|, q != p, Q be a q-Sylow subgroup of G. Prove that Q acts on B by conjugation. 2. Let Q' be the image of Q in G/H, show that if |Q'|=|Q| then KxB is (isomorphic to) a subgroup of G, where K is a kernel of the action in (1). 3. Suppose Q is normal in G, does that imply that the image of KxB in G/H is normal in G/H? 2. Relevant equations The standard theory of the Sylow theorems. In the previous parts of the question morphism here helped me prove that G=HN_G(B) where N_G(B) is the normalizer of B in G, and that G/H is isomorphic to N_G(B)/N_H(B) 3. The attempt at a solution (1) Actually, I think there must be some mistake in the question. For Q to act on B we need that Q be a subset of the normalizer of B in G, and that doesn't seem to be the case. It might be that I'm missing something here... (2) This would imply that the intersection of Q and H is trivial, but I'm not sure how K would "look like" to gain intuition on how this could unfold. (3) I think not, but if I could present G as some direct product that might give me a reason why.