The Herzog-Sch\"onheim Conjecture for small groups and harmonic subgroups

TL;DR

This paper proves the Herzog-Schönheim Conjecture for all groups smaller than order 1440, showing that in any non-trivial coset partition, some subgroups have equal indices, and explores harmonic properties of subgroup indices with pairwise trivial cosets.

Contribution

It establishes the conjecture for small groups and links subgroup indices with harmonic integers in specific coset configurations.

Findings

Herzog-Schönheim Conjecture holds for groups of order less than 1440.

In certain coset partitions, some subgroups have equal indices.

Subgroups with pairwise trivial cosets have harmonic indices when n ≤ 4.

Abstract

We prove that the Herzog-Sch\"onheim Conjecture holds for any group $G$ of order smaller than $1440$. In other words we show that in any non-trivial coset partition $\{g_i U_i\}_{i=1}^n $ of $G$ there exist distinct $1 \leq i, j \leq n$ such that $[G:U_i]=[G:U_j]$. We also study interaction between the indices of subgroups having cosets with pairwise trivial intersection and harmonic integers. We prove that if $U_1$,...,$U_n$ are subgroups of $G$ which have pairwise trivially intersecting cosets and $n \leq 4$ then $[G:U_1]$,...,$[G:U_n]$ are harmonic integers.