The mod 2 cohomology algebra of a Sylow 2-subgroup of the general linear group \(GL(4,{\mathbb{Z}}_ 2)\)
For \(n\geq 2\), let \(GL_{n,2}({\mathbb{Z}}/2)\) denote the 2-Sylow subgroup of \(GL_ n({\mathbb{Z}}/2)\). In the paper under review the mod 2 cohomology algebra of \(GL_{4,2}({\mathbb{Z}}/2)\) is computed by means of the cohomology spectral sequence of the central extension \[ 0\to Z\to GL_{4,2}({\mathbb{Z}}/2)\to GL_{4,2}({\mathbb{Z}}/2)/Z\to 1, \] where \(Z\cong {\mathbb{Z}}/2\) denotes the centre of \(GL_{4,2}({\mathbb{Z}}/2)\). The basic tool is modular invariant theory. The group \(GL_{4,2}({\mathbb{Z}}/2)/Z\) is nilpotent of class 2, and it admits an obvious analogue D(n,2) in each dimension \(n\geq 4\) which is a certain factor group of \(GL_{n+1,2}({\mathbb{Z}}/2)\) and again nilpotent of class 2. In an appendix the mod 2 cohomology of the groups D(n,2) is given. \{Reviewer's remark: A related paper by \textit{R. J. Milgram} and \textit{S. Priddy} [J. Pure Appl. Algebra 44, 291-302 (1987)] is not among the references.\}
scientific article; zbMATH DE number 3989640