K^*(BG) rings for groups $G=G_{38},...,G_{41}$ of order 32

TL;DR

This paper extends the understanding of Morava K-theory rings for specific groups of order 32, showing they are generated by transferred Euler classes and explicitly computing their ring structures.

Contribution

It generalizes previous results to arbitrary s and provides explicit descriptions of the ring structures for these groups.

Findings

$K(s)^*(BG)$ is generated by transferred Euler classes.

The ring $K(s)^*(BG)$ is a quotient of a polynomial ring in 6 variables.

Explicit generators for the defining ideal are provided.

Abstract

B. Schuster \cite{SCH1} proved that the $mod$ 2 Morava $K$-theory $K(s)^*(BG)$ is evenly generated for all groups $G$ of order 32. For the four groups $G$ with the numbers 38, 39, 40 and 41 in the Hall-Senior list \cite{H}, the ring $K(2)^*(BG)$ has been shown to be generated as a $K(2)^*$-module by transferred Euler classes. In this paper, we show this for arbitrary $s$ and compute the ring structure of $K(s)^*(BG)$. Namely, we show that $K(s)^*(BG)$ is the quotient of a polynomial ring in 6 variables over $K(s)^*(pt)$ by an ideal for which we list explicit generators.