Group automorphisms inducing the identity map on cohomology. (English) Zbl 0624.20024

Let G be a group and R a commutative ring. Denote by Aut(G;R) resp. \(Aut_ u(G;R)\) the subgroup of Aut(G) consisting of those automorphisms which induce the identity on the cohomology of G with coefficients in R resp. which are restrictions of inner automorphisms of RG. It is shown that Inn(G)\(\leq Aut_ u(G;R)\leq Aut(G;R)\), and \(Inn(G)=Aut_ u(G;{\mathbb{Z}})\) if G is finite with a normal Sylow 2-subgroup; \(Aut_ u(G;{\mathbb{Z}})/Inn(G)\) is an elementary abelian 2-group for arbitrary G. Using complex representation and the Atiyah spectral sequence the authors show that Aut(G;\({\mathbb{Q}})\) preserves all conjugacy classes, if G is a connected Lie group, which fails to be true in general. If G is finite solvable, \(| Aut(G;{\mathbb{Z}})|\) divides a power of \(| G|\).
Reviewer: W.Grölz


20E36 Automorphisms of infinite groups
20G10 Cohomology theory for linear algebraic groups
20J05 Homological methods in group theory
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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