Herzog–Schönheim conjecture

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 conjecturedthat 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

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

:

are pairwise disjoint with the indices distinct, but they do not cover multiples of $2^k$.

In 2004 Zhi-Wei Sun proved an extended versionof the Herzog-Schönheim conjecture in the case where $G\_1,ldots,G\_k$ are subnormal in $G$ [http://arxiv.org/abs/math/0306099] . A basic lemma in Sun's proof states that if $G\_1,ldots,G\_k$ are subnormal and of finite index in $G$, then

:

and hence

:

where $P(n)$ denotes the set of prime
divisors of $n$.

ee also

* Covering system