## Hiroshima Mathematical Journal

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

### 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**

Received: 26 April 2017

Revised: 14 June 2019

First available in Project Euclid: 7 March 2020

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.32917/hmj/1583550012

**Mathematical Reviews number (MathSciNet)**

MR4074376

**Zentralblatt MATH identifier**

07197867

**Subjects**

Primary: 55N20: Generalized (extraordinary) homology and cohomology theories

Secondary: 55R12: Transfer 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20]

**Keywords**

Morava $K$-theory Euler class transfer

#### 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