Herzog–Schönheim conjecture

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In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory.

Let $G$ be a group, and let

$A=\{a_1G_1,\ \ldots,\ a_kG_k\}$

be a finite system of left cosets of subgroups $G_1,\ldots,G_k$ of $G$ .

In 1974 M. Herzog and J. Schönheim conjectured that if $A$ forms a partition of $G$ with $k > 1$ , then the (finite) indices $[G:G_1],\ldots,[G:G_k]$ cannot be distinct. In contrast, if $H$ is a subgroup of $G$ with index $k=[G:H]<\infty$ then $G$ can be partitioned into $k$ left cosets of $H$ .

When $G$ is the additive group $\Z$ of integers, this is a conjecture of Paul Erdős confirmed by H. Davenport, L. Mirsky, D. Newman and R. Rado independently. Observe that

$2^0+2\Z,\ 2+2^2\Z,\ \ldots,\ 2^{k-1}+2^k\Z$

are pairwise disjoint with the indices $[\Z:2\Z],\ldots,[\Z:2^k\Z]$ distinct, but they do not cover multiples of $2 k$ .

In 2004 Zhi-Wei Sun proved an extended version of the Herzog-Schönheim conjecture in the case where $G_1,\ldots,G_k$ are subnormal in $G$ [1]. A basic lemma in Sun's proof states that if $G_1,\ldots,G_k$ are subnormal and of finite index in $G$ , then

$\bigg[G:\bigcap_{i=1}^kG_i\bigg]\ \bigg|\ \prod_{i=1}^k[G:G_i]$