Two groups $(X_1,\,*_1)$ and $(X_2,\,*_2)$ are said to be isomorphic if there is a group isomorphism $\psi\colon X_1\to X_2$

Next we name a few necessary conditions for two groups $X_1,\,X_2$ to be isomorphic (with isomorphism $\psi$ as above).

1. If two groups are isomorphic, then they have the same cardinality. Indeed, an isomorphism is in particular a bijection of sets.
2. If the group $X_1$ has an element $g$ of order $n$ then the group $X_2$ must have an element of the same order. If there is an isomorphism $\psi$ then $\psi(g)\in X_2$ and $(\psi(g))^n=\psi(g^n)=\psi(e_1)=e_2$ where $e_i$ is the identity elements of $X_i$ Moreover, if $(\psi(g))^m=e_2$ then $\psi(g^m)=e_2$ and by the injectivity of $\psi$ we must have $g^m=e_1$ so $n$ divides $m$ Therefore the order of $\psi(g)$ is $n$
3. If one group is cyclic, the other one must be cyclic too. Suppose $X_1$ is cyclic generated by an element $g$ Then it is easy to see that $X_2$ is generated by the element $\psi(g)$ Also if $X_1$ is finitely generated, then $X_2$ is finitely generated as well.
4. If one group is abelian, the other one must be abelian as well. Indeed, suppose $X_2$ is abelian. Then $$\psi(g*_1 h)=\psi(g)*_2 \psi(h)=\psi(h)*_2 \psi(g) =\psi(h*_1 g)$$ and using the injectivity of $\psi$ we conclude $g*_1 h=h*_1 g$

Note. Isomorphic groups are sometimes said to be abstractly identical, because their ``abstract'' structures are completely similar -- one may think that their elements are the same but have only different names.