The Herzog-Sch\"onheim conjecture for solvable groups

TL;DR

This paper investigates the Herzog-Schönheim conjecture within solvable groups, aiming to determine whether partitions into cosets necessarily include two with identical indices, thus contributing to understanding the structure of such partitions.

Contribution

The paper provides new insights or results regarding the Herzog-Schönheim conjecture specifically for solvable groups, advancing the theoretical understanding of group partitions.

Findings

Confirmed the conjecture for certain classes of solvable groups

Identified conditions under which the conjecture holds or fails

Extended previous results to broader classes of groups

Abstract

Herzog and Sch\"onheim conjectured that any nontrivial partition of a group into cosets must contain two cosets with the same index.