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Chern approximations for generalised group cohomology. (English summary) Topology 40 (2001), no. 6, 1167–1216. This paper is devoted to the study of E0BG, the E-cohomology of the classifying space of a finite group G. There are strong hypotheses on E; the main examples that satisfy the hypotheses are (an extension of) Morava K-theory K(n) and Morava E-theory En. The main idea of the paper is to construct the closest possible approximation C(E,G) to E0BG using only the complex representation theory ofG (but not transfers). There is a natural map C(E,G) → E0BG. This approximation is an isomorphism when G is abelian or when E is p-adic complex K-theory and G is a p-group. This was essentially known before, but the author also works out detailed examples when G is Σ3, Σ4, and Q8, where the approximation is also an isomorphism. But it can’t be an isomorphism in general, as the author shows by examining extraspecial p-groups for p> 2. The approximation C(E,G) is always a finitely generated E0-module, and the author constructs a character theory for it analogous to the Hopkins-Kuhn-Ravenel character theory for E0BG [M. J. Hopkins, N. J. Kuhn and D. C. Ravenel, J. Amer. Math. Soc