A likely obvious question concerning sufficient conditions for (sub)group equality.

Question

Is the following true?

Let $G$ be a finite group, with subgroups $H$ and $K$. Then, $H \le K$ and $|H| = |K|$ if and only if $H = K$.

It seems like it most certainly should be true, but I'm actually having trouble proving the "$ \implies$" direction.

Answer 1

Since $G$ is finite, we know $H$ and $K$ must be finite. If $H$ is a subgroup of $K$, then it is, in particular, a subset of $K$. So, if in addition $|H|=|K|$, it must follow that $H=K$.