complete group
A complete group is a group $G$ that is

1.
centerless (center $Z(G)$ of $G$ is the trivial group), and

2.
any of its automorphism^{} $g:G\to G$ is an inner automorphism^{}.
If a group $G$ is complete^{}, then its group of automorphisms,
$\mathrm{Aut}(G)$, is isomorphic to $G$. Here’s a quick
proof. Define $\varphi :G\to \mathrm{Aut}(G)$ by
$\varphi (g)={g}^{\mathrm{\#}}$, where ${g}^{\mathrm{\#}}(x)=gx{g}^{1}$. For $g,h\in G$,
${(gh)}^{\mathrm{\#}}(x)=(gh)x{(gh)}^{1}=g(hx{h}^{1}){g}^{1}=({g}^{\mathrm{\#}}{h}^{\mathrm{\#}})(x)$,
so $\varphi $ is a homomorphism^{}. It is onto because every $\alpha \in \mathrm{Aut}(G)$ is inner, (=${g}^{\mathrm{\#}}$ for some $g\in G$). Finally,
if ${g}^{\mathrm{\#}}(x)={h}^{\mathrm{\#}}(x)$, then $gx{g}^{1}=hx{h}^{1}$, which means
$({h}^{1}g)x=x({h}^{1}g)$, for all $x\in G$. This implies that
${h}^{1}g\in Z(G)=\u27e8e\u27e9$, or $h=g$. $\varphi $ is
onetoone.
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 

Canonical name  CompleteGroup 
Date of creation  20130322 15:21:46 
Last modified on  20130322 15:21:46 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  5 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 20E36 
Classification  msc 20F28 