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 automorphismPlanetmathPlanetmathPlanetmathPlanetmath g:GG is an inner automorphismMathworldPlanetmath.

If a group G is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, then its group of automorphisms, Aut(G), is isomorphic to G. Here’s a quick proof. Define ϕ:GAut(G) by ϕ(g)=g#, where g#(x)=gxg-1. For g,hG, (gh)#(x)=(gh)x(gh)-1=g(hxh-1)g-1=(g#h#)(x), so ϕ is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmath. It is onto because every αAut(G) is inner, (=g# for some gG). Finally, if g#(x)=h#(x), then gxg-1=hxh-1, which means (h-1g)x=x(h-1g), for all xG. This implies that h-1gZ(G)=e, or h=g. ϕ is one-to-one.

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


  • 1 J. Rotman, The Theory of Groups, An Introduction, Allyn and Bacon, Boston (1965).
Title complete group
Canonical name CompleteGroup
Date of creation 2013-03-22 15:21:46
Last modified on 2013-03-22 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