IsomorphicGroups 2003-10-15 01:28:57 2006-01-11 07:40:15 Definition isomorphic abstractly identical \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{isomorphic} Two groups $(X_1,\,*_1)$ and $(X_2,\,*_2)$ are said to be {\em 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). \begin{enumerate} \item {\it If two groups are isomorphic, then they have the same cardinality}. Indeed, an isomorphism is in particular a bijection of sets. \item {\it 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 $n$. \item {\it 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 {\it if $X_1$ is finitely generated, then $X_2$ is finitely generated as well}. \item {\it 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 \,$g*_1 h=h*_1 g$. \end{enumerate} \textbf{Note.}\, Isomorphic groups are sometimes said to be {\em abstractly identical}, because their ``abstract'' \PMlinkescapetext{structures} are completely similar --- one may think that their elements are the same but have only different names.