I've just started self-learning group theory, and I came accros Cayley's theorem. Which made me wondering about the following question: Given two groups $G$ and $H$, and there exist an injective homomorphism $f: G \rightarrow H$, then $G$ is isomorphic to a subgroup of $H$.
My attempt: Let $K$ be the range of the homomorphism. Then $f: G \rightarrow K$ is surjective, but it is also injective, so bijective. So $G$ and $K$ are isomorphic. But how do I know that $K$ is a subgroup? I think it is enough that since $K$ is isomorphic to a group $G$, $K$ must be a group, but why a subgroup of $H$. Sure, the set of $K$ is a subset of the set of $G$, but this not a proof. Can someone give the correct argument?
Best Answer
For the neutral element: $e_H = f(e_G) \in K$.
For the inverse: for $y \in K$, let $x \in G$ with $f(x) = y$. Then $y^{-1} = f(x)^{-1} = f(x^{-1}) \in K$.
For the product: let $y, y' \in K$. Let $x, x' \in G$ with $f(x) = y, \space f(x') = y'$. Then $y y' = f(x) f(x') = f(x x') \in K$.
These are the three properties that characterize a subgroup.