Paul
Posts:
676
Registered:
7/12/10


Re: name for definition in group theory
Posted:
Mar 24, 2013 5:34 PM


On Sunday, March 24, 2013 9:03:01 PM UTC, Ken Pledger wrote: > In article <m89uk8lshngii7afp7qktplg90ubnq9doj@4ax.com>, > > David C. Ullrich <ullrich@math.okstate.edu> wrote: > > > > > On Sun, 24 Mar 2013 08:15:15 0700 (PDT), Paul <pepstein5@gmail.com> > > > wrote: > > > > > > >Does anyone know the name for the following property of a group G: G has > > > >no nontrivial 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 nonAbelian, then it has nontrivial 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 nonobvious that the proof holds without AC, but I wasn't thinking about the axiom of choice until Jose raised it.
Paul Epstein

