The Herzog-Schönheim Conjecture for small groups and harmonic subgroups

Abstract

We prove that the Herzog-Schönheim 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 $1\le i,j\le n$ 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 $n\le 4$ then $[G:U_1]$,...,$[G:U_n]$ are harmonic integers.

16 pages