On Sun's conjecture concerning disjoint cosets

TL;DR

This paper proves Sun's conjecture for the case of three and four pairwise disjoint cosets in a group, establishing a lower bound on the gcd of their indices.

Contribution

The paper confirms Sun's conjecture for k=3 and k=4, advancing understanding of the structure of disjoint cosets in groups.

Findings

Confirmed Sun's conjecture for k=3

Confirmed Sun's conjecture for k=4

Established gcd lower bounds for these cases

Abstract

In 2004, Zhi-Wei Sun posed the following conjecture: If a_1G_1,...,a_kG_k (k>1) are finitely many pairwise disjoint left cosets in a group G with all the indices [G:G_i] finite, then for some 1\le i<j\le k, the greatest common divisor of [G:G_i] and [G:G_j] is at least k. In this paper, we confirm Sun's conjecture for k=3,4.