Order of $\langle f(g)\rangle$ If $f$ is a homomorphism from $G$ to $H,$ if $g\in G$,
in general do $\langle g\rangle$ and $\langle f(g)\rangle$ have the same order? I feel like they can have different orders, but I can't think of a counterexample...

It's a good question.
Actually all that are needed is a few very basic ingredients (more basic than Lagrange and the first isomorphism theorem):

*

*$x^n=e\implies \lvert x\rvert \mid n$ (this may be the trickiest,  you need the division algorithm)

*$h$ a homomorphism $\implies h(x^n)=h(x)^n$

*$x^{\lvert x\rvert}=e$

*$h$ a homomorphism $\implies h(e)=e$ (follows from the second bullet btw).

Now try putting these together to prove $$\lvert h(x)\rvert \mid \lvert x\rvert $$.
In general, we don't get equality (whenever the kernel of $h$ is non-trivial).
