On the restriction map in the \(\text{mod-}2\) cohomology of groups
Let \(G\) be a 2-group, \(H^*(G)\) the mod-2 cohomology of \(G\) and \(\text{Res}_G\) the restriction map from \(H^*(G)\) to all proper subgroups of \(G\). This paper investigates the nilpotence degree of the ideal \(\text{Ker Res}_G\). In [J. Pure Appl. Algebra 67, No. 1, 33-37 (1990; Zbl 0787.20030)] \textit{T. Marx} showed that \(\text{Ker Res}_G\) is of nilpotence degree 2, provided that \(G\) is generated by two elements, or the Frattini subgroup of \(G\) is central and is not elementary abelian. Here the author proves that the same holds for 3-generator groups \(G\) satisfying one of the following conditions (i) \(|G|\leq 2^6\); (ii) the Frattini subgroup of \(G\) is central; (iii) \(G\) is an extension of either an abelian group, or \(D_8\times C_2\), or \(Q_8\times C_2\). Moreover, the author obtains another proof of T. Marx's result. \textit{G. S. Avrunin} and \textit{J. F. Carlson} [in Proc. Am. Math. Soc. 118, No. 2, 339-343 (1993; Zbl 0788.20029)] and \textit{K. Inoue} and \textit{A. Kuno} [in J. Math. Kyoto Univ. 33, No. 4, 1047-1055 (1993; Zbl 0809.20041)] have also studied the nilpotence degree of elements of \(H^*(G)\).
scientific article; zbMATH DE number 810225