Let $G$ and $H$ be groups. If $G \cong H$, then the following invariants agree for $G$ and $H$:
paragraph
admin
1. the cardinality of the underlying set, so $|G| = |H|$;
2. abelianness, so $G$ is abelian if and only if $H$ is abelian;
3. cyclicity, so $G$ is cyclic if and only if $H$ is cyclic;
4. for every $n \in \mathbb{N}$, the cardinalities of the sets of elements of order $n$ agree:
list
admin
\begin{align*}
\left|\{g \in G : \operatorname{ord}(g) = n\}\right| = \left|\{h \in H : \operatorname{ord}(h) = n\}\right|.
\end{align*}