Herzog-Schönheim Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a subgroup $H\le G$, the index $[G:H]$ is the number of its left cosets. A nontrivial finite coset partition is a disjoint union \[ G=\bigsqcup_{i=1}^{s} g_iH_i,\qquad s\ge2, \] where each $g_i\in G$ and each $H_i<G$ is a subgroup of finite index. There is no assumption that $G$ is finite, abelian or finitely generated. The nontriviality requirement excludes the one-part partition $G=G$. Right-coset and left-coset formulations are equivalent by inversion.

Statement recorded in the supplied source.

For every group $G$ and every nontrivial finite coset partition $G=\bigsqcup_{i=1}^{s}g_iH_i$, must there be distinct indices $i,j\in\{1,\ldots,s\}$ such that \[ [G:H_i]=[G:H_j]? \] [S2]

Short English statement

In every nontrivial partition of a group into finitely many subgroup cosets, must two of the subgroups have the same index?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1110 (ALG-010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Fabienne Chouraqui, The Herzog--Schönheim conjecture for finitely generated groups, arXiv:1803.08301 (2018), Section 1, including the formulation for arbitrary groups. https://arxiv.org/abs/1803.08301.