Proof that a group is finite

Question

$(G,.)$ is a group and $H$ a subgroup of it so that $G-H$ is finite. Prove that G is finite.

I found the next proof:

$f_a:H \to G-H ,f_a(x)=ax, a\in G-H$ . Since $f_a$ is injective and $G-H$ is finite it results that H is finite. $G=H \cup(G-H)$ so $G$ is finite.

Can somebody explain me how it results that $H$ is finite, please?

Answer 1

Hint: The cosets of $H$ partition $G$.

Answer 2

Well, $H$ needs to be a proper subgroup of $G$ first of all.

Let $a$ be any element in $G - H$. Then $|aH| = |H|$ and $aH \subseteq G \setminus H$ [make sure you see why to both of these]. So if $G \setminus H$ is finite then so is $aH$ and (as $|aH| = |H|$) thus so is $H$.

Answer 3

in general, if there is an injective fuction from A to a finite set, then A is also a finite set.