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 quickproof. 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$h^{-1}g\in Z(G)=⟨e⟩$, or $h=g$. $\varphi$ isone-to-one.

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