CMI 2023 | Mathematics | Part B | Question: 2

Question

(A) (3 marks) Let $G$ be a group such that $|G|=p^{a} d$ with $a \geq 1$ and $(p, d)=1$. Let $P$ be a Sylow $p$-subgroup and let $Q$ be any $p$-subgroup of $G$ such that $Q$ is not a subgroup of $P$. Show that $P Q$ is not a subgroup of $G$.

(B) (7 marks) Let $\Gamma$ be a group that is the direct product of its Sylow subgroups. Show that every subgroup of $\Gamma$ also satisfies the same property.