Direct product and Sylow subgroups Let $G$ be a finite group that is equal to inner direct product of its subgroup $P$ and $Q$, where $P$ is a Sylow $p$-subgroup and $Q$ is a Sylow $q$-subgroup of $G$. If $L \le G$, prove that $L$ is equal to the inner direct product of $P \cap L$ and $Q \cap L$.

The order of $G$ is $p^mq^n$ for some integers $m,n$. $L$, being a subgroup of $G$, has order $p^aq^b$ for some $a$ and $b$. $P$, being a normal Sylow subgroup, contains every subgroup of order a power of $p$, and similarly for $Q$. Thus $P\cap L$ is a normal Sylow $p$-subgroup of $L$ and $Q\cap L$ is a normal Sylow $q$-subgroup of $L$. $P\cap L$ and $Q\cap L$ intersect trivially because $P$ and $Q$ do and by cardinality must generate $L$, hence $L$ is the direct product of $P\cap L$ and $Q\cap L$.
