The Wells exact sequence for the automorphism group of a group extension

Abstract

We obtain an explicit description of the Wells map for the automorphism group of a group extension in the full generality and investigate the dependency of this map on group extensions. Some applications are given.Crown

Extensions and automorphisms of Lie algebras

“…Here, we consider the abelian additive group structure on Z 1 (B; A). The sequence is similar to the one derived for extensions of groups by Wells in [13] and studied subsequently in [8,9,10,11]. Since s and λ are F -linear maps, it follows that γ λ is also F -linear.…”

“…As an application, they obtained the structure of the automorphism group of free metabelian Lie algebra L/L (2) of finite rank, and showed that it has automorphisms which cannot be lifted to automorphisms of L. However, to our knowledge, almost nothing seems to be known about Problem 1. For extensions of groups, the analogous problem has been investigated recently in [8,9,10,11], wherein using cohomological methods the problem for finite groups has been reduced to p-groups.…”

On the cohomology of finite soluble groups

“…Here the final map is not a homomorphism, although it is in fact a derivation of Q-modules-see [4]. From the Wells sequence and Theorems 1 and 3, we deduce immediately: Theorem 5.…”

Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms

“…Therefore, we obtain a map t : C ψ → Aut h ((g ⋉ h) T ⊕S ), (β, α) → t (β,α) . The map t obviously satisfies τ t = id C ψ which implies that t is a section of the map τ in the exact sequence (20). Further, the map t is a group homomorphism.…”

“…Another interesting study related to extensions of algebraic structures is given by the inducibility of pair of automorphisms. See [20,25,26,30] for more details about this problem in (abelian) group extensions. In the context of Lie algebras, the problem can be described as follows.…”