Automorphisms of group extensions

Abstract

If 1 ->■ G -í> E^-Tt -> 1 is a group extension, with i an inclusion, any automorphism <j> of E which takes G onto itself induces automorphisms t on G and a on n.However, for a pair (a, t) of automorphism of n and G, there may not be an automorphism of E inducing the pair.Let à: n -*■ Out G be the homomorphism induced by the given extension.A pair (a, t) e Aut n x Aut G is called compatible if a fixes ker á, and the automorphism induced by a on Hü is the same as that induced by the inner automorphism of Out G determined by t.Let C< Aut IT x Aut G be the group of compatible pairs.Let Aut (E; G) denote the group of automorphisms of E fixing G.The main result of this paper is the construction of an exact sequence 1 -» Z&T1, ZG) -* Aut (E; G)-+C^ H*(l~l, ZG).The last map is not surjective in general.It is not even a group homomorphism, but the sequence is nevertheless "exact" at C in the obvious sense.

Locations

• Transactions of the American Mathematical Society - View - PDF