# complete group

A complete group is a group $G$ that is

1. 1.

centerless (center $Z(G)$ of $G$ is the trivial group), and

2. 2.

any of its automorphism $g\colon G\to G$ is an inner automorphism.

If a group $G$ is complete, then its group of automorphisms, $\operatorname{Aut}(G)$, is isomorphic to $G$. Here’s a quick proof. Define $\phi\colon G\to\operatorname{Aut}(G)$ by $\phi(g)=g^{\#}$, where $g^{\#}(x)=gxg^{-1}$. For $g,h\in G$, $(gh)^{\#}(x)=(gh)x(gh)^{-1}=g(hxh^{-1})g^{-1}=(g^{\#}h^{\#})(x)$, so $\phi$ is a homomorphism. It is onto because every $\alpha\in\operatorname{Aut}(G)$ is inner, (=$g^{\#}$ for some $g\in G$). Finally, if $g^{\#}(x)=h^{\#}(x)$, then $gxg^{-1}=hxh^{-1}$, which means $(h^{-1}g)x=x(h^{-1}g)$, for all $x\in G$. This implies that $h^{-1}g\in Z(G)=\langle e\rangle$, or $h=g$. $\phi$ is one-to-one.

It can be shown that all symmetric groups on $n$ letters are complete groups, except when $n=2$ and $6$.

## References

• 1 J. Rotman, The Theory of Groups, An Introduction, Allyn and Bacon, Boston (1965).
Title complete group CompleteGroup 2013-03-22 15:21:46 2013-03-22 15:21:46 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 20E36 msc 20F28