Proof verification: The function $f: H \to gH$ shows that $|H|=|gH|$, the number of elements in subgroup is equal to number of coset elements.

Question

Can someone please verify my proof?

Let $H$ be a subgroup of a group $G$ and let $g \in G$. Define a function $f: H \to gH$ by $f(h) = gh$. Since this map is bijective, $|H| = |gH|$.

Proof:

Is it okay that I upload the proof as a photo?

Answer 1

The proof looks fine, besides that you go from the letter "$g$" that is used in the statement of the problem to "$b$", for some reason.

For pedagogical reasons, I would state the second part of the proof differently: I would not say, "for all $bh \in bH$", but I would do something like how one shows that two sets are identical: If I want to show that the range of $f$ and $bH$ are identical as sets, I take an arbitrary element of the codomain ($bH$), give it a name, then show that it is in the range of $f$. Showing that the range of $f$ is in $bH$ would show that $f$ is well-defined, which might be something that might be required in the problem. (I would recommend adding it into the proof, because it is part of showing that $|H| = |bH|$.)