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Herzog–Schönheim conjecture for solvable groups

The Herzog–Schönheim question for solvable groups asks whether any exact partition of a finite or infinite solvable group into at least two left cosets must use two subgroups with the same index.

Why it matters and what counts as progress

Why it matters: It asks whether solvability alone forces repeated subgroup indices in every exact coset partition, even for infinite groups.

What counts as progress: Prove the repeated-index conclusion for all solvable groups, including infinite groups, or produce an exact coset partition with pairwise distinct indices.

How the problem developed

  1. 1974 · Conjecture [1]

    M. Herzog and J. Schönheim Proposed that every finite coset partition of a group contains two cosets whose subgroups have equal index.

  2. September 1986 · Special case [2]

    Marc A. Berger, Alexander Felzenbaum and Aviezri Fraenkel Proved the conjecture for finite nilpotent groups, a solvable-group subclass.

  3. 30 Sept 2025 · Special case [3]

    Martino Garonzi and Leo Margolis Reported proofs for finite simple and symmetric groups; these families do not cover all solvable groups.

  4. 2 Oct 2026 · Solvable group result [4]

    Murali Menon Registered a Lean-verified index-form proof for every solvable group, finite or infinite; the arbitrary-group conjecture remains open.

AI contribution

Sources

  1. Jonoska, Savchuk and Chouraqui, An Approach to the Herzog-Schönheim Conjecture Using Automata
  2. Berger, Felzenbaum and Fraenkel, The Herzog-Schönheim Conjecture for Finite Nilpotent Groups
  3. Martino Garonzi and Leo Margolis, The Herzog-Schönheim conjecture for simple and symmetric groups
  4. Murali Menon, The Herzog–Schönheim conjecture for solvable groups — Palomar registry
  5. Menon’s solvable-group formalization
  6. Erdős Problem 274 formal statement
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