I am looking at a proof regarding if $\alpha : G \to H$ is a homomorphism, the order of $\alpha(g)$ divides the order of $g$, for $g \in G$.
So I let $|g| = n$.
$\alpha(g)^n = \alpha(g^n) = \alpha(1_G) = 1_H$
So $n$ is a multiple of $|\alpha(g)|$.
That makes sense but what I am not sure about is how we are allowed to bring the exponent $n$ inside the bracket...$\alpha(g)^n = \alpha(g^n)$
We are saying performing some binary operation on the image of $g$ $n$ times is the same as performing the binary operation on $g$ $n$ times and then taking the image of it. Why are we allowed to say this?
Edit: Would someone mind showing me how it would work for $g^3$ or $g^4$ $\ldots$ as it is exponents greater than 2 that are giving me problems.