The Herzog–Schönheim conjecture for small groups and harmonic subgroups

Abstract

We prove that the Herzog–Schonheim 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 $$U_1,\ldots ,U_n$$ are subgroups of G which have pairwise trivially intersecting cosets and $$n \le 4$$ then $$[G:U_1],\ldots ,[G:U_n]$$ are harmonic integers.