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arxiv: 2606.29866 · v1 · pith:JH57O2NFnew · submitted 2026-06-29 · 🧮 math.GR

Characterization of the alternating and symmetric groups by the order and conjugacy class sizes

Ilya Gorshkov , Andrey V. Vasil'ev This is my paper

Pith reviewed 2026-06-30 04:12 UTC · model grok-4.3

classification 🧮 math.GR
keywords finite groupsconjugacy class sizesalternating groupssymmetric groupssimple groupsgroup characterizationisomorphism
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The pith

Any finite group with the same order and same set of conjugacy class sizes as an alternating or symmetric group must be isomorphic to it.

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

The paper establishes that order combined with the set of conjugacy class sizes forms a complete invariant for alternating and symmetric groups. If a finite group G matches an alternating or symmetric group S on these two quantities, then G is isomorphic to S. The argument for these families is then joined to earlier results on other simple groups to conclude that every simple group is likewise determined by its order and conjugacy class sizes. A reader would care because these quantities are among the most elementary numerical data attached to a group and are often easier to compute than the full multiplication table.

Core claim

We prove that an arbitrary finite group G having the same order and same set of conjugacy class sizes as an alternating or symmetric group S must be isomorphic to S. From this and previously known results it follows that the same holds true for every simple group S.

What carries the argument

The pair consisting of the group order and the set of sizes of its conjugacy classes.

If this is right

  • Alternating and symmetric groups are uniquely recoverable from their order and conjugacy class size set.
  • The same uniqueness holds for every finite simple group once the alternating and symmetric cases are included.
  • Two finite simple groups that agree on order and on the set of conjugacy class sizes must be isomorphic.

Where Pith is reading between the lines

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

  • The result supplies a numerical test that could be used in computational group recognition when only class-size data is available.
  • It raises the question whether the same pair of invariants determines other families of groups, such as solvable groups or groups of Lie type in small rank.
  • One could test the boundary by checking whether the invariants distinguish non-simple groups that happen to share order and class sizes with a simple group.

Load-bearing premise

The previously published characterizations of the remaining simple groups by the same invariants are correct.

What would settle it

Exhibit a finite group G that is not isomorphic to a given alternating or symmetric group S yet satisfies |G| = |S| and possesses exactly the same set of conjugacy class sizes.

read the original abstract

We prove that an arbitrary finite group $G$ having the same order and same set of conjugacy class sizes as an alternating or symmetric group $S$ must be isomorphic to $S$. From this and previously known results it follows that the same holds true for every simple group $S$.

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 paper proves that if a finite group G has the same order and the same set of conjugacy class sizes as an alternating or symmetric group S, then G is isomorphic to S. Combining this with previously established results for other simple groups, the characterization extends to all finite simple groups.

Significance. If correct, the result completes a characterization of all finite simple groups by order and conjugacy class sizes, a natural extension of prior work on recognition problems in finite group theory. The direct argument for A_n and S_n avoids reliance on the classification of finite simple groups beyond the invoked prior characterizations.

minor comments (3)
  1. The abstract states the main theorem but provides no indication of the proof strategy or key lemmas; a brief outline in the introduction would improve accessibility.
  2. Notation for conjugacy class sizes (e.g., the set cs(G)) should be defined explicitly in §1 rather than assumed from prior literature.
  3. The extension to all simple groups in the final paragraph relies on citations; ensure the references include the exact statements being invoked for each family of simple groups.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. The report correctly notes that the result, combined with prior characterizations, completes the recognition of all finite simple groups by order and conjugacy class sizes, using only the invoked earlier results rather than the full CFSG.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct proof that any finite group G with the same order and conjugacy class sizes as an alternating or symmetric group S is isomorphic to S. It then notes that the result for all simple groups follows from this plus previously known results for the remaining cases. No step in the provided abstract or claim structure reduces a prediction or central result to a fitted parameter, self-definition, or load-bearing self-citation chain within this manuscript. The reliance on external prior characterizations is explicit and does not create internal circularity, as those results are treated as independent inputs rather than derived from the current argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all background group-theoretic facts are treated as standard.

pith-pipeline@v0.9.1-grok · 5566 in / 893 out tokens · 37490 ms · 2026-06-30T04:12:53.327850+00:00 · methodology

discussion (0)

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Reference graph

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22 extracted references · 1 linked inside Pith

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