In a homework problem I was doing, I was trying to show that $U(8)$ is not isomorphic to $U(10)$. They used that, supposing $f: U(10) \rightarrow U(8)$ was an isomorphism, $|f(3)| = |3| = 4$ and there are no elements of order $4$ in $U(8)$, thus disproving that any such isomorphism could exist.
I am confused. For a given homomorphism $\varphi : G \rightarrow H$, I understand it is true that $|\varphi(x)|$ divides $|x|$, for some $x\in G$.
For an isomorphism $\varphi : G \rightarrow H$, is it the case that $|\varphi(x)| = |x|$, for $x\in G$?
If not, can you explain the reasoning in the proof given at the beginning?
It is the case that for an isomorphism $\phi:G\to H$, $|\phi(x)|=|x|$ for $x\in G$.