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.
Title  isomorphic groups 

Canonical name  IsomorphicGroups 
Date of creation  20130322 14:01:58 
Last modified on  20130322 14:01:58 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  10 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 20A05 
Defines  isomorphic 
Defines  abstractly identical 