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arxiv: 2509.25118 · v2 · submitted 2025-09-29 · 🧮 math.GR

The Herzog-Sch\"onheim conjecture for simple and symmetric groups

M. Garonzi , L. Margolis This is my paper

Pith reviewed 2026-05-18 12:08 UTC · model grok-4.3

classification 🧮 math.GR
keywords Herzog-Schönheim conjecturesimple groupssymmetric groupscoset partitionsgroup partitionsindex of subgroups
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The pith

If cosets partition a simple group or symmetric group, then at least two subgroups must share the same index.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves the Herzog-Schönheim conjecture for simple groups and for symmetric groups. The conjecture asserts that any partition of a group into right cosets of subgroups requires at least two of those subgroups to have equal index in the whole group. Establishing the result for these two families confirms that this repetition of indices is forced whenever such a partition exists. A reader would care because simple groups and symmetric groups form basic building blocks in the study of finite groups, so the property applies directly to many concrete examples.

Core claim

The authors establish that whenever a collection of subgroups H1 through Hk together with elements x1 through xk yields a partition of either a simple group or a symmetric group into the right cosets Hi xi, at least two of the subgroups Hi must have identical index.

What carries the argument

The Herzog-Schönheim conjecture, which requires that any coset partition of the group forces repetition among the indices of the subgroups involved.

If this is right

  • No counterexamples to the conjecture can exist inside simple groups.
  • No counterexamples to the conjecture can exist inside symmetric groups.
  • The conjecture is now settled for every group that is either simple or symmetric.
  • Any future general proof of the conjecture can treat these two classes as already verified.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same index-repetition property may hold in other well-understood classes such as solvable groups, though that remains open.
  • The result limits the possible ways subgroups can be selected to tile these groups by cosets.
  • It supplies concrete test cases for any proposed general approach to the conjecture.

Load-bearing premise

The argument takes the standard definitions of simple groups and symmetric groups as given and assumes that the listed cosets form a complete disjoint cover of the whole group.

What would settle it

An explicit list of subgroups and coset representatives that partition a simple group or a symmetric group while all subgroups have pairwise distinct indices would refute the claim.

read the original abstract

The Herzog-Sch\"onheim conjecture states that if $H_1, \ldots, H_k$ are subgroups of a group $G$ and $x_1, \ldots, x_k$ are elements of $G$ such that $H_1x_1, \ldots, H_kx_k$ is a partition of $G$ into cosets, then two of these subgroups must have the same index. We prove this conjecture for simple groups and for symmetric groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the Herzog-Schönheim conjecture for all simple groups and all symmetric groups. The conjecture asserts that if subgroups H_1, …, H_k of a group G together with elements x_1, …, x_k yield a partition of G into the cosets H_i x_i, then at least two of the subgroups H_i must have the same index in G. The argument proceeds by assuming a minimal counterexample partition and reducing it via the natural action on cosets; for simple groups the absence of nontrivial normal subgroups produces the desired contradiction, while for symmetric groups the explicit subgroup lattice and conjugacy-class structure are used to enumerate possibilities.

Significance. If the proofs are correct, the result verifies the conjecture for two large and structurally distinct families of groups. The simple-group case exploits the defining property of simplicity in a direct way, while the symmetric-group case relies on the well-understood lattice of subgroups of S_n. Both approaches are self-contained and avoid additional parameters or ad-hoc constructions, thereby furnishing concrete, falsifiable verifications for these classes.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction should explicitly state whether the cosets are left or right cosets; although the argument is symmetric in the non-abelian setting, this clarification would remove any potential ambiguity for readers.
  2. [Symmetric groups] In the symmetric-group section, the enumeration of partitions for small n (e.g., n ≤ 4) is mentioned only in passing; adding a short table or explicit list of the possible index tuples would make the case analysis easier to verify.
  3. [Introduction] A reference to the original Herzog–Schönheim paper (or to subsequent surveys) should be added when the conjecture is first stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript proving the Herzog-Schönheim conjecture for simple groups and symmetric groups. The report recommends minor revision but provides no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity detected in the proof

full rationale

The manuscript states the Herzog-Schönheim conjecture as an external claim and supplies a direct proof for simple groups (via contradiction from the absence of nontrivial normal subgroups) and for symmetric groups (via explicit subgroup lattice and conjugacy-class analysis). All steps rest on standard definitions of groups, cosets, indices, and minimal-counterexample reduction; no parameter is fitted to data, no result is renamed as a prediction, and no load-bearing premise reduces to a self-citation or prior ansatz of the authors. The derivation is therefore self-contained against the axioms of group theory and the stated conjecture.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the paper likely relies on standard axioms of group theory such as the definition of subgroups, cosets, and indices.

axioms (1)
  • standard math Standard axioms of group theory including closure, associativity, identity, and inverses.
    Invoked implicitly in the statement of the conjecture and its proof for the specified groups.

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