Suppose that $G_1$ and $G_2$ are finite groups and $\beta: G_1\to G_2$ is an isomorphism. If $x_2 = \beta(x_1)$ for a given element $x_1 \in G_1$, prove that $x_1$ and $x_2$ have the same order.
I think I should use induction but I am not sure how could I prove it?
You don't need induction: saying $x_1$ has order $n$ means there's a isomorphism $\mathbf Z/n\mathbf Z\longrightarrow \langle x_1\rangle$ (mapping $1$ to $x_1$). Just left-compose this isomorphism with $\beta$ to obtain an isomorphism $\mathbf Z/n\mathbf Z\longrightarrow \langle x_2\rangle$, mapping $1$ to $x_2$.