On the first Zassenhaus conjecture and direct products

TL;DR

This paper investigates the behavior of the first Zassenhaus conjecture and the General Bovdi Problem under direct products, establishing new cases where these conjectures hold, especially for Sylow tower and Frobenius groups.

Contribution

It extends the validity of (ZC1) and (Gen-BP) to new classes of groups, including Sylow-by-abelian, Sylow tower, and certain Frobenius groups, using an extended HeLP method.

Findings

(Gen-BP) holds for Sylow tower groups and supersolvable groups.

(ZC1) is established for direct products of Sylow-by-abelian groups under certain conditions.

(ZC1) and (Gen-BP) hold for direct products involving Frobenius groups and finite abelian groups.

Abstract

In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products as well as the General Bovdi Problem (Gen-BP) which turns out to be a slightly weaker variant of (ZC1). Among others we prove that (Gen-BP) holds for Sylow tower groups, so in particular for the class of supersolvable groups. (ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group. We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G \times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.