If two groups are isomorphic, does this imply that they have the same number of elements with the same order?

Question

If two groups are isomorphic, does this imply that they have the same number of elements with the same order? For example, if a group $G$ is isomorphic to a group $H$, and $G$ has $2$ elements of order $3$, I know that $H$ must have element(s) of order $3$, but does it have to be exactly $2$ elements of order $3$?

I am trying to prove that $S_4$ and $SL_2(\mathbb{Z}_3)$ are not isomorphic, and I found that $S_4$ has $9$ elements of order $2$ and $SL_2(\mathbb{Z}_3)$ has only $1$ element of order $2$, but I am not sure if that’s enough to prove that they’re not isomorphic.

Answer 1

The group $SL(2,3)$ has center $$ \{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 2 & 0 \\ 0 & 2 \\\end{pmatrix} \} $$ of order $2$, whereas $S_4$ has trivial center. Hence the two groups cannot be isomorphic. This argument might be easier here than counting the number of elements with a given order.