Herzog-Schonheim Conjecture

The Herzog-Schonheim conjecture, in group theory, is a combinatorial problem posed by Marcel Herzog and Jochanan Schonheim in 1974. It considers a finite system of left cosets of subgroups of a group that together form a partition of the group into more than one part, and conjectures that the indices of those subgroups cannot then all be distinct. By contrast, when repeated indices are allowed, partitioning a group into cosets is easy: any subgroup of finite index k partitions the group into k left cosets of itself. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/