On Sunday, March 24, 2013 9:03:01 PM UTC, Ken Pledger wrote: > In article <email@example.com>, > > David C. Ullrich <firstname.lastname@example.org> wrote: > > > > > On Sun, 24 Mar 2013 08:15:15 -0700 (PDT), Paul <email@example.com> > > > wrote: > > > > > > >Does anyone know the name for the following property of a group G: G has > > > >no non-trivial automorphisms. ? > > > >Thank you > > > > > > These groups are referred to as "groups of order 1 or 2". > > > > > > There must be a very elementary proof of this. I know > > > no group theory; here's a not quite elementary proof > > > using a big result from harmonic analysis.... > > > > > > If G is non-Abelian, then it has non-trivial inner automorphisms > > x -> (a^(-1))xa. Therefore it's Abelian, so x -> x^(-1) is an > > automorphism, etc., as you showed. >
Indeed, but this seems to be exactly what David said in his second posting. Of course, this doesn't quite complete the proof but David completed the argument. I find it non-obvious that the proof holds without AC, but I wasn't thinking about the axiom of choice until Jose raised it.