Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 9 / a / Solution

Both subgroups contain the identity element, so $H\cap K$ is nonempty. If $x,y\in H\cap K$, then $xy\in H$ and $xy\in K$, so $xy\in H\cap K$. Similarly $x^{-1}$ belongs to both subgroups and hence to their intersection. Associativity is inherited from $G$. ThusThis verifies the group axioms; finiteness is unnecessary.