## Hiroshima Mathematical Journal

### All extensions of $C_2$ by $C_{2^n} \times C_{2^n}$ are good for the Morava $K$-theory

#### Abstract

Let $C_m$ be a cyclic group of order $m$. We prove that if a group $G$ fits into an extension $1 \rightarrow C^2_{2^{n+1}} \rightarrow G \rightarrow C_2 \rightarrow 1$ for $n\geq 1$ then $G$ is good in the sense of Hopkins-Kuhn-Ravenel, i.e., $K(s)^\ast(BG)$ is evenly generated by transfers of Euler classes of complex representations of subgroups of $G$.

#### Note

The author is supported by Shota Rustaveli National Science Foundation Grant 217-614 and CNRS PICS Grant 7736.

#### Article information

Source
Hiroshima Math. J., Volume 50, Number 1 (2020), 1-15.

Dates
Revised: 14 June 2019
First available in Project Euclid: 7 March 2020

https://projecteuclid.org/euclid.hmj/1583550012

Digital Object Identifier
doi:10.32917/hmj/1583550012

Mathematical Reviews number (MathSciNet)
MR4074376

Zentralblatt MATH identifier
07197867

#### Citation

Bakuradze, Malkhaz. All extensions of $C_2$ by $C_{2^n} \times C_{2^n}$ are good for the Morava $K$-theory. Hiroshima Math. J. 50 (2020), no. 1, 1--15. doi:10.32917/hmj/1583550012. https://projecteuclid.org/euclid.hmj/1583550012