Hochschild-Serre spectral sequences of split extensions and cohomology of some extra-special \(p\)-groups
Let \(p\) be a prime, \(G\) an extra special \(p\)-group and consider the extensions \[ (1)\qquad 1\to Z\to G\to G/Z\to 1\qquad\text{and}\qquad(2)\qquad 1\to A\to G\to G/A\to 1, \] where \(Z\) is the center and \(A\) a maximal subgroup of \(G\). The Hochschild-Serre spectral sequences of both of these extensions converge to the associated graded rings \(H^*(G)=H^*(G,F_p)\). The first one has been used by several authors and it seems to have the disadvantage that its spectral sequence converges at a high term. By using the Charlap-Vasquez theorem, \textit{S. F. Siegel} [J. Pure Appl. Algebra 106, No. 2, 185-198 (1996; Zbl 0861.20050)] has used the second one and proved that for \(p>2\) and \(G\cong E(p^3)\), the extra special \(p\)-group of order \(p^3\) and exponent \(p\), the spectral sequence converges at \(E_2\) for \(p=3\) and at \(E_3\) for \(p>3\). In this paper, the author also analyzes the Hochschild-Serre spectral sequence of the extension (2), in the cases where \(A\) can be chosen such that (2) is split. He proves that \(E_2=E_\infty\) for \(p=2\) and this gives an alternative proof of \textit{D. Quillen}'s theorem on the determination of \(H^*(G)\) [Math. Ann. 194, 197-212 (1971; Zbl 0225.55015)]. He also solves the spectral sequence in the case when \(p>2\) and \(G\cong E(p^3)\) by using the image of the restriction \(\text{Res}^G_A\) and the fact that \(d_r\) commutes with some automorphisms of \(G\). Although these results are not original they show the effectiveness of the spectral sequence of (2) for the computation of \(H^*(G)\).
scientific article; zbMATH DE number 1308224
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