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

MSC:

 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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