# isomorphic groups

Two groups $(X_{1},\,*_{1})$ and $(X_{2},\,*_{2})$ are said to be 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. 1.

If two groups are isomorphic, then they have the same cardinality. Indeed, an isomorphism is in particular a bijection of sets.

2. 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. 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. 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.

Title isomorphic groups IsomorphicGroups 2013-03-22 14:01:58 2013-03-22 14:01:58 alozano (2414) alozano (2414) 10 alozano (2414) Definition msc 20A05 isomorphic abstractly identical