On the IYB-property in some solvable groups

TL;DR

This paper explores the properties of IYB (Involutive Yang-Baxter) groups, especially in solvable and nilpotent groups, providing new constructions and insights into their structure and relation to the isomorphism problem.

Contribution

It demonstrates that certain semidirect products of nilpotent groups of class two with coprime IYB groups are IYB, including Hertweck's counterexample, and analyzes the concrete IYB structure of this example.

Findings

Semidirect products of nilpotent class two groups with coprime IYB groups are IYB.

Hertweck's counterexample to the isomorphism conjecture is an IYB group.

Concrete IYB structures on Hertweck's counterexample are characterized.

Abstract

A finite group $G$ is called Involutive Yang-Baxter (IYB) if there exists a bijective 1-cocycle $\chi: G \longrightarrow M$ for some $\mathbb Z G$-module $M$. It is known that every IYB-group is solvable, but it is still an open question whether the converse holds. A characterization of the IYB property by the existence of an ideal $I$ in the augmentation ideal $\omega \mathbb Z G$ complementing the set $1-G$ lead to some speculation that there might be a connection with the isomorphism problem for $\mathbb Z G$. In this paper we show that if $N$ is a nilpotent group of class two and $H$ is an IYB-group of order coprime to that of $N$, then $N\rtimes H$ is IYB. The class of groups that can be obtained in that way (and hence are IYB) contains in particular Hertweck's famous counterexample to the isomorphism conjecture as well as all of its subgroups. We then investigate what an IYB structure on Hertweck's counterexample looks like concretely.