Morava K-theory rings of the extensions of C_2 by the products of cyclic 2-groups

TL;DR

This paper computes the Morava K-theory rings of certain extensions of cyclic 2-groups, expanding understanding of their algebraic structure for specific groups of order 32.

Contribution

It extends previous work by explicitly calculating the Morava K-theory rings for new groups, using a combination of known theorems and explicit algebraic methods.

Findings

Explicit ring structures for G_{36}, G_{37}, G_{34}, G_{35} determined.

Morava K-theory rings are quotients of polynomial rings with explicit generators.

Complexity of rings similar across different group types.

Abstract

In \cite{SCH1} Schuster proved that $mod$ 2 Morava $K$-theory $K(s)^*(BG)$ is evenly generated for all groups $G$ of order 32. There exist 51 non-isomorphic groups of order 32. In \cite{H}, these groups are numbered by $1, \cdots ,51$. For the groups $G_{38},\cdots, G_{41}$, that fit in the title, the explicit ring structure is determined in \cite{BJ}. In particular, $K(s)^*(BG)$ is the quotient of a polynomial ring in 6 variables over $K(s)^*(pt)$ by an ideal generated by explicit polynomials. In this article we present some calculations using the same arguments in combination with a theorem of \cite{B0} on good groups in the sense of Hopkins-Kuhn-Ravenel. In particular, we consider the groups $G_{36},G_{37}$, each isomorphic to a semidirect product $(C_4\times C_2\times C_2)\rtimes C_2$, the group $G_{34}\cong (C_4\times C_4)\rtimes C_2$ and its non-split version $G_{35}$. For these groups the action of $C_2$ is diagonal, i.e., simpler than for the groups $G_{38},\cdots, G_{41}$, however the rings $K(s)^*(BG)$ have the same complexity.