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→G is an inner automorphism
.
If a group G is complete, then its group of automorphisms,
Aut(G), is isomorphic to G. Here’s a quick
proof. Define ϕ:G→Aut(G) by
ϕ(g)=g#, where g#(x)=gxg-1. For g,h∈G,
(gh)#(x)=(gh)x(gh)-1=g(hxh-1)g-1=(g#h#)(x),
so ϕ is a homomorphism
. It is onto because every α∈Aut(G) is inner, (=g# for some g∈G). Finally,
if g#(x)=h#(x), then gxg-1=hxh-1, which means
(h-1g)x=x(h-1g), for all x∈G. This implies that
h-1g∈Z(G)=⟨e⟩, or h=g. ϕ 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 |
---|---|
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 |