Suppose $\varphi:G_1\to G_2$ is a homomorphism and $H$ is the kernel of $\varphi$. We want to show that if $G_1$, $G_2$ are finite, then $G_1/H$ is finite.
My question is mainly about what can we say about the order of group $G'$ and $G''$ if there is a bijection/injection/surjection from $G'$ to $G''$? Is it analogous to the set theory?
For this specific question, my attempt is as follows:
since $G_1/H$ is the quotient group of $G_1$, $o(G_1/H)\mid o(G_1)$. Since there is a bijection between $G_1/H$ and $\varphi(G_1)$ and $\varphi(G_1)\subseteq G_2$, there is an injection from $\varphi(G_1/H)$ to $G_2$. $\textbf{Then I was not sure if I can assert that}$ $o(G_1/H)\mid o(G_2)$. If so, $o(G_1,H)$ is a common factor of $o(G_1)$ and $o(G_2)$ hence finite.
thank you