Journal of Pure and Applied Algebra 44 (1987) 241-250 North-Holland 241 zyxwvutsrq GROUP AUTOMORPHISMS INDUCING THE IDENTITY MAP ON COHOMOLOGY Stefan JACKOWSKI and Zbigniew MARCINIAK Instytut Matematyki, Uniwersytet Warszawski, PKiN IXp. PL 00-901 Warszawa, Poland Communicated by E.M. Friedlander and S. Priddy Received 4 September 1985 1. Statement of results Let G be a topological group and let R be a commutative ring. The automor- phisms of G which induce the identity homomorphism on cohomology with coeffi- cients in R of the classifying space BG form a subgroup Aut*(G; R) of the group of all automorphisms Aut(G). In this paper we investigate how this subgroup is related to other, better known, subgroups of the automorphism group. It is well known that the group of inner automorphisms Inn(G) is a subgroup of Aut*(G; R). More generally, we define a group Aut,(G; R) 2 Inn(G) consisting of the automorphisms of G which are restrictions of those inner automorphisms of the group ring RG which normalize the subset G c RG. In Section 2 we show that Aut,.,(G; R) c Aut*(G; R). Little is known about Autu(G; R). Let Z be the ring of integers. For finite p- groups Coleman [4] has proved that Aut&G; Z) = Inn(G). In Section 3 we prove that the same is true for all finite groups with normal Sylow 2-subgroups. For an arbitrary group G we show that Aut,(G; Z)/Inn(G) is an elementary abelian 2-group. In other words, any automorphism from Autu(G; Z) performed twice becomes inner. We did not find any example of a group G for which Inn(G)# Aut,(G; 77). In Section 3 we also prove that Aut,(G; R) is contained in the group Aut,(G) consisting of the automorphisms preserving conjugacy classes. If R is a field and charR{IGl, then Aut&G; R)=Aut,(G) and Aut*(G; R)=Aut(G). We conclude the paper by discussing some relations between the groups Aut*(G; R) and Au&(G). Such relations are suggested by the Atiyah spectral se- quence [l]: for a compact Lie group G all automorphisms from Aut*(G; Z) act trivially on the graded ring associated with complex representation ring R(G). On the other hand Aut,(G) consists of those automorphisms which act trivially on R(G) itself. In this context the conjecture: Aut*(G; Z) c Aut,(G) arises in a natural way. Using characteristic classes of complex representations we show in Section 4 0022-4049/87/$3.50 0 1987, Elsevier Science Publishers B.V. (North-Holland)