isomorphic groups

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

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 $X_{\mathrm{1}}$ has an element $g$ of order $n$, then the group $X_{\mathrm{2}}$ 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 $X_{\mathrm{1}}$ is finitely generated, then $X_{\mathrm{2}}$ 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” are completely similar — one may think that their elements are the same but have only different names.