```Date: Mar 24, 2013 12:25 PM
Author: David C. Ullrich
Subject: Re: name for definition in group theory

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 non-trivial automorphisms.    ?   >Thank youThese groups are referred to as "groups of order 1 or 2".There must be a very elementary proof of this. I knowno group theory; here's a not quite elementary proofusing a big result from harmonic analysis:A topological group is a group together with a topologysuch that the group operations are continuous.A (continuous) character of a topological group Gis a continuous homomorphism of G into the unitcircle in the complex plane. If G is a topologicalgroup then the set of continuous characters isdenoted G^*; note that G^* is itself a group,with multiplication defined pointwise.Now, if G is a locally compact abelian (LCA) groupthen the Pontryagin Duality Theorem states thatG is isomorphic to its second dual (G^*)^*.That's the non-trivial part.(Oops, there's a missing definition there. If G isa LCA group then there is a natural topology onthe group G^*; it turns out that G^* is also LCA.)Ok. Assume G is a group with no non-trivialautomorphisms. Since all the inner automophismsof G are trivial, G must be abelian.Give  G the discrete topology. Now G is an LCAgroup. Let K = G^*. (K is compact, not thatwe need that here.) Then G is isomorphic toK^*.Now, if chi is a character of K then chi^*, thecomplex conjugate, is also a character of K.The map chi -> chi^* is an automorphism ofK^*. This automorphism must be trivial, soevery chi in K^* must be real-valued.So every chi in K^* takes only the values 1 and -1.Hence every non-trivial element of K^* has order 2.So. G is an abelian group and every non-trivial elementof G has order 2. This means that G is a vector spaceover the field Z_2 = {0,1}.If dim(G) = 0 or 1 then |G| = 1 or 2. If dim(G) > 1 thenG has a non-trivial automorphism.The elementary proof would start by noting that G mustbe abelian, as above, and then invoke some structuretheorem or something to deduce that every elementhas order 2, since any cyclic group of order greaterthan 2 has an automorphism...>>Paul Epstein
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