On finite groups whose order is divisible by two primes.

Question

Let $G$ be a finite group whose order is divisible by $p$ and $q$ only. Suppose $P \in Syl_p(G)$ such that $P$ is a normal subgroup. Does this imply that a Sylow $q$-subgroup of $G$ is isomorphic to $\frac{G}{P}$? Thank you for your input!

Answer 1

If $Q$ is any $q$-Sylow subgroup, then $QP$ is a subgroup of $G$ since $P$ is normal, and by cardinality considerations $QP=G$, since $G$ only has two distinct prime divisors. Then $$G/P=QP/P\simeq Q/(Q\cap P)\simeq Q$$ by the (second) isomorphism theorem, and since $Q\cap P$ is trivial. An explicit isomorphism could also be found as $Q\to G/P:q\mapsto qP$.