What is the possible value for the subgroup of index 2?

Question

Let $H\leq G$ and $[G:H]\leq 2$. If the $|G|=n$ then $|H|=n$ if $n$ is odd. What is happening when $n$ is even? What are the possible values for $|H|$ when $n$ is even?

Thanks.

Answer 1

If $[G:H] \leq 2$ and $|G|=2k$ then $|H| = k$ or $|H| =2k$. If $|H|=k$ then $[G:H] = 2$ and H is a normal subgroup that contains every square. If $|H|=2k$ then $H=G$.