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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
Full Text: DOI
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