The \(K(n)\)-Euler characteristic of extraspecial \(p\)-groups
type: Item
instance of: scholarly article
title: The \(K(n)\)-Euler characteristic of extraspecial \(p\)-groups
zbMATH Open document ID: 0964.55007
DOI: 10.1016/S0022-4049(99)00102-4
author: Maurizio Brunetti
published in: Journal of Pure and Applied Algebra
publication date: 2001-07-13T00:00:00Z
review text: Let \(E\) be a finite \(p\)-group which occurs in a central extension \(1 \to N \to E \to V \to 1\) where \(N\) is cyclic of order \(p\) and \(V\) is an elementary abelian group of order \(p^{2m}\). \(E\) is an extraspecial group if \(Z(E)\) is of order \(p\). The cohomology of such groups is known if \(p = 2\) [\textit{D. Quillen}, Math. Ann. 194, 197-212 (1971; Zbl 0225.55015)] but not for odd primes. The paper under review calculates the Morava \(K\)-theory Euler characteristic, \(\chi=\text{rank}(K(n)^{\text{even}}(BE)) - \text{rank} (K(n)^{\text{odd}} (BE))\), of such a group for \(p\) odd, giving a computable formula in terms of the number of \(B\)-isotropic subspaces of an \({\mathbb F}_p\) vector space where \(B\) is a nondegenerate alternating form. The author suggests that this result will lead to information about the mod \(p\) cohomology of \(E\) via the Atiyah-Hirzebruch spectral sequence.
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 1563006
zbMATH Keywords: extraspecial p-group
zbMATH Keywords: Morava \(K\)-theory
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1563006