Timeline for A condition on finite groups

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Jan 31, 2012 at 19:33	comment	added	Marty Isaacs		It is interesting that the set of "invisible" automorphisms forms an abelian subgroup of Aut(G). To be more precise: Let H be any subgroup of G and let A be a group of automorphisms of G fixing each element of H and each right coset of H in G. Then [G,A] is contained in H, so [G,A,A] = 1. Obviously then, [A,G,A] = 1. By the three-subgroups lemma, [A,A,G] = 1. Thus means that the derived subgroup A' = [A,A] acts trivially on G. Since we are working with automorphisms, this says that A' = 1, so A is abelian. There is much more about this sort of thing in Section 4C of my group theory book.
May 30, 2010 at 21:34	history	answered	Torsten Ekedahl	CC BY-SA 2.5