Recognition by element orders for simple linear and unitary groups

Maria A. Grechkoseeva , Alexey M. Staroletov , Andrey V. Vasil'ev

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Pith reviewed 2026-05-10 19:46 UTC · model grok-4.3

classification 🧮 math.GR

keywords element ordersrecognition problemsimple linear groupssimple unitary groupsfinite simple groupsgroup spectrum

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The pith

The recognition problem for finite simple linear and unitary groups is now fully solved using their sets of element orders.

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

This paper finishes determining the number h(G) of non-isomorphic finite groups that share the same set of element orders as each finite simple linear or unitary group G. It handles the remaining open cases after earlier results covered smaller ranks and fields. A reader would care because this determines exactly when these groups can be identified uniquely from element orders alone and lists any exceptions that share orders with them. The work uses the classification of finite simple groups to rule out other possibilities and relies on prior recognition results for the cases not treated here.

Core claim

For every finite simple linear or unitary group G, the value of h(G) is now known and every finite group H with the same set of element orders as G is explicitly described, completing the recognition problem for these families.

What carries the argument

The set ω(G) of element orders (the spectrum) of a finite group G, which is used to count and classify all groups sharing the same spectrum as the given simple group.

Load-bearing premise

That the cases already treated in prior papers together with the results here cover every possibility and that the classification of finite simple groups is correct.

What would settle it

Existence of a finite group H that is not among the groups listed in this or prior papers yet has exactly the same element orders as some simple linear or unitary group.

read the original abstract

For a finite group $G$, let $\omega(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $\omega(H)=\omega(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if $h(G)$ is finite, then all finite groups $H$ with $\omega(H)=\omega(G)$ are described. We complete the solution of the recognition problem for the finite simple linear and unitary 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 completes the solution of the recognition problem for the finite simple linear groups PSL(n,q) and unitary groups PSU(n,q). For each such G, it determines the value of h(G), the number of pairwise non-isomorphic finite groups H with the same set of element orders ω(H)=ω(G), and explicitly describes all such H whenever h(G) is finite. The argument proceeds by partitioning the parameter space (n,q) into cases already treated in the literature and the remaining open cases, which are resolved via direct computation of element orders together with reductions that invoke the classification of finite simple groups and previously established recognition results for smaller-rank or smaller-field groups.

Significance. If the results hold, the work has substantial significance: it finishes a long-running program on recognition by element orders for the two largest families of classical simple groups. The approach is systematic, relying on exhaustive case division rather than a single uniform argument, and the explicit appeal to prior results for the bulk of the parameter space makes the completeness claim verifiable in principle. The paper supplies the final pieces needed to know h(G) for every PSL(n,q) and PSU(n,q).

minor comments (3)

[Introduction] Introduction, paragraph 3: the list of previously solved cases for linear and unitary groups would be clearer if presented as a compact table rather than a narrative paragraph.
[§4.3] §4.3, the statement of the main theorem for PSU(n,q) with n≥4: the reduction step invokes a 2018 result on element orders without restating the precise hypotheses under which that result applies; a one-sentence reminder of the conditions would improve readability.
[Table 2] Table 2 (element-order sets for small q): the column headers use the same font size as the body text, making the table harder to scan; increasing header weight or adding light shading would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for recommending minor revision. The work completes the recognition problem by element orders for all finite simple linear groups PSL(n,q) and unitary groups PSU(n,q), determining h(G) and describing the groups H with the same element orders when h(G) is finite. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript completes the recognition problem for PSL(n,q) and PSU(n,q) by supplying explicit case-by-case proofs for the remaining parameter ranges, using direct computation of element orders together with the external Classification of Finite Simple Groups for the global reduction and previously published independent results for smaller-rank or smaller-field groups. No equation or theorem in the paper reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims rest on new direct arguments that are externally falsifiable and do not rename or smuggle prior ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a proof completion within the classification of finite simple groups and the recognition-by-element-orders program; it therefore inherits the standard axioms of finite group theory and the correctness of earlier papers in the same series.

axioms (1)

domain assumption Classification of finite simple groups (CFSG)
Recognition results for simple groups of Lie type routinely invoke the CFSG to reduce to known lists of groups.

pith-pipeline@v0.9.0 · 5396 in / 1204 out tokens · 119486 ms · 2026-05-10T19:46:22.302845+00:00 · methodology

discussion (0)

Reference graph

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