A complete group is a group that is
centerless (center of is the trivial group), and
If a group is complete, then its group of automorphisms,
, is isomorphic to . Here’s a quick
proof. Define by
, where . For ,
so is a homomorphism. It is onto because every is inner, (= for some ). Finally,
if , then , which means
, for all . This implies that
, or . is
It can be shown that all symmetric groups on letters are complete groups, except when and .
- 1 J. Rotman, The Theory of Groups, An Introduction, Allyn and Bacon, Boston (1965).