Herzog–Schönheim conjecture

In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974.

Let ${\displaystyle G}$ be a group, and let

${\displaystyle A=\{a_{1}G_{1},\ \ldots ,\ a_{k}G_{k}\}}$

be a finite system of left cosets of subgroups ${\displaystyle G_{1},\ldots ,G_{k}}$ of ${\displaystyle G}$.

Herzog and Schönheim conjectured that if ${\displaystyle A}$ forms a partition of ${\displaystyle G}$ with ${\displaystyle k>1}$, then the (finite) indices ${\displaystyle [G:G_{1}],\ldots ,[G:G_{k}]}$ cannot be distinct. In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if ${\displaystyle H}$ is any subgroup of ${\displaystyle G}$ with index ${\displaystyle k=[G:H]<\infty }$ then ${\displaystyle G}$ can be partitioned into ${\displaystyle k}$ left cosets of ${\displaystyle H}$.