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.
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.