All extensions of $C_2$ by $C_{2^{n+1}}\times C_{2^{n+1}}$ are good

TL;DR

This paper proves that all group extensions of the form $C_2$ by $C_{2^{n+1}} imes C_{2^{n+1}}$ are 'good' in the sense of Hopkins-Kuhn-Ravenel, meaning their cohomology is generated by transfers of Euler classes, extending previous results.

Contribution

It generalizes the known case for $n=1$ to all $n$, establishing that such extensions are 'good' in the specified cohomological sense.

Findings

All such extensions are 'good' in the Hopkins-Kuhn-Ravenel sense.

The cohomology $K(s)^*(BG)$ is evenly generated by transfers of Euler classes.

Extension property holds for all $n$, not just $n=1$.

Abstract

Let $C_{m} $ be a cyclic group of order $m$. We prove that if the group $G$ fits into an extension $1\to C_{2^{n+1}}^2\to G\to C_2\to 1$ then $G$ is good in the sense of Hopkins-Kuhn-Ravenel, i.e., $K(s)^*(BG)$ is evenly generated by transfers of Euler classes of complex representations of subgroups of $G$. Previously this fact was known for $n=1$.