Subject: [math-fun] Groups all of whose automorphisms are inner Message-ID: <1400633545.44072.YahooMailNeo@web162102.mail.bf1.yahoo.com> Content-Type: text/plain; charset=iso-8859-1
Are there theorems that characterize groups all of whose automorphisms are inner?? For example, for a finite group, the characters of a matrix representation must be algebraic integers, and the presence of a non-rational integer in the character table betrays the existence of an outer automorphism. ? --? Gene
--is the group An (n!/2 even permutations of n items) a counterexample? It has an outer aut (conjugation with an odd perm), but at least for A3=C3 all characters are ordinary integers, hence no "betrayal" of the outer automorphism. C4 is another counterexample. So your test is (assuming valid) only a 1-sided test. However, it might be that there are not very many such counterexamples, and hence after some enumerated list of exceptions are excluded then your test might become a genuine test. This hopefully could be proven using the Classification Theorem, i.e. all finite simple groups and their character tables are known. All finite groups arise by "gluing together" simple groups. If we could argue that, there is an outer aut exactly when the glued simples include at least one with an outer aut, and if we further could argue about the character tables... then maybe prove something. (This is all speculative, I'm raving here.)