On a Noether normalization for a \(\text{mod-}2\) cohomology ring (Q1906624)

A 2-group \(\widetilde{V}\) is called almost extraspecial if it is a central extension of an elementary abelian group \(V\) by \(\mathbb{Z}/2\). Then \(\widetilde{V}\) is characterized by the quadratic form \(Q:V\to\mathbb{Z}/2\) induced by squaring. Here, \(Q\) is assumed to be nonsingular. Let \(H^*(\widetilde{V})\) be the mod-2 cohomology ring of \(\widetilde{V}\) and \(\Delta\) a unique faithful irreducible real representation of \(\widetilde{V}\). The ring structure of \(H^*(\widetilde{V})\) was completely determined by \textit{D. Quillen} [in Math. Ann. 194, 197-212 (1971; Zbl 0225.55015)]. Let \(I(\widetilde{V})\) be the subring of universally stable elements in \(H^*(\widetilde{V})\) defined by \textit{L. Evens} and \textit{S. Priddy} [in Q. J. Math., Oxf. II. Ser. 40, No. 160, 399-407 (1989; Zbl 0687.20047)] \(I(\widetilde{V})=\bigcap_G\text{Im(res}:H^*(G)\to H^*(\widetilde{V}))\) where \(G\) runs over all finite groups with \(\widetilde {V}\) as a Sylow 2-subgroup. Then [in Tokyo J. Math. 15, No. 1, 91-97 (1992; Zbl 0777.20020)] the author showed that, except when \(\widetilde{V}\) is the dihedral group of order 8, \(I(\widetilde{V})\) is the invariant subring of the subgroup of the orthogonal group stabilizing \(Q\), which is generated by the elements of odd order. The main goal of this paper is to show that \(I(\widetilde{V})\) is very close to the subring generated by Stiefel-Whitney classes of \(\Delta\) , which is known as a Noether normalization for \(H^*(\widetilde{V})\). The main theorem shows that, stating explicitly what happens in each case: real, complex and quaternion.