On a Noether Normalization for a mod-2 Cohomology Ring

Abstract

An almost extraspecial group with a nonsingular quadratic form is treated. There are three cases: real, complex, and quaternion. The mod-2 cohomology ring has three subrings: the subring generated by Stiefel-Whitney classes of a unique faithful irreducible real representation, an invariant subring of the orthogonal group, and that of its commutator subgroup. The object of this paper is to show that the three subrings are very close to one another. Particularly in the quaternion case, they coincide with each other module the nilradical of the cohomology ring. It is well known that the first is a Noether normalization for the cohomology ring, while it was shown by us that the third is a subring of universally stable elements defined by Evens and Priddy (with a few exceptions of small order). Thus it may be concluded that the last is nothing but a Noether normalization for the cohomology ring.