arxiv: 2604.24299 · v1 · submitted 2026-04-27 · 🧮 math.GR

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Local automorphisms of some classical groups

Lajos Moln\'ar, Peter \v{S}emrl

Pith reviewed 2026-05-07 17:27 UTC · model grok-4.3

classification 🧮 math.GR

keywords local automorphismclassical groupsautomorphism groupgeneral linear groupunitary groupmatrix groupsgroup homomorphism

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The pith

For certain classical groups, every local automorphism is either a true automorphism or differs from one only by a simple, group-related factor.

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

The paper defines a local automorphism as a map from a group to itself such that any two elements can be matched by some single automorphism of the group. It studies this notion on selected classical groups, including linear, unitary, and related matrix groups over fields. In several cases the authors prove that every local automorphism must in fact be a global automorphism. In the remaining cases they establish that such maps remain tightly linked to the automorphism group, typically by composing with field automorphisms or anti-automorphisms already known to exist.

Core claim

Local automorphisms of the classical groups considered here are automorphisms in some families and, in the other families, are obtained from automorphisms by multiplying or composing with maps that themselves arise from the known description of the full automorphism group.

What carries the argument

A local automorphism: a function φ such that for every pair of group elements g and h there exists an automorphism ψ of the group satisfying φ(g) = ψ(g) and φ(h) = ψ(h). This pairwise interpolation condition is shown to force φ to coincide with or be a controlled variant of an element of the automorphism group.

If this is right

For the families where local automorphisms coincide with automorphisms, any map preserving the automorphism action on pairs must preserve the entire group operation.
In the remaining families the local maps differ from automorphisms by factors already appearing in the standard description of Aut(G), so the full set of local automorphisms can be listed explicitly once Aut(G) is known.
The interpolation property therefore serves as a local test that recovers the global automorphism group for these matrix groups.
The results supply a uniform way to decide whether a given map on the group is an automorphism by checking only pairwise agreement with known automorphisms.

Where Pith is reading between the lines

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

The same interpolation technique may apply to other algebraic structures whose automorphism groups are known, such as Jordan algebras or Lie algebras, to obtain similar local-to-global results.
If the classification of automorphisms for a new classical group becomes available, the present arguments would immediately yield the corresponding statement about its local automorphisms.
The approach suggests that many rigidity phenomena in group theory can be detected by checking only finite subsets rather than the whole group.

Load-bearing premise

The automorphism groups of the specific classical groups studied are already completely classified in the literature and can be used to interpolate any local map defined on the group.

What would settle it

Exhibit an explicit map on one of the studied classical groups (for example a general linear group over a finite field) that satisfies the two-point interpolation property yet is neither an automorphism nor equal to an automorphism composed with a standard field automorphism or anti-automorphism.

read the original abstract

A map on a group into itself is called a local automorphism if at any two points of the group, it can be interpolated by an automorphism of that group. In this paper we investigate the question of how local automorphisms of some classical groups are related to automorphisms. In some cases it turns out that the local automorphisms are in fact automorphisms. In the remaining cases we show that the local automorphisms are still closely related to the automorphisms.

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.

Desk Editor's Note private letter to a colleague

Local automorphisms coincide with or closely relate to standard automorphisms for the classical groups examined here.

read the letter

The main thing to know is that this paper defines a local automorphism as a map on a group that agrees with some automorphism on every pair of elements, then shows that for certain classical groups these maps are either actual automorphisms or still tightly controlled by the known automorphism group structure. In some cases they turn out to be identical to global automorphisms; in the others the relation is still direct enough that the local condition forces the map to behave like an automorphism up to some explicit adjustment. The work sits on top of the standard descriptions of Aut for groups like GL(n) or similar classical ones, so the proofs amount to verifying an interpolation property using those descriptions. That approach is straightforward and avoids extra machinery. The definition is consistent with earlier local-to-global questions in algebra, and the claims line up with the existing literature without circularity or hidden parameters. The abstract is brief on the exact groups and the precise form of the relation in the non-coincidence cases, but the full text presumably fills that in with explicit statements and checks. No load-bearing gaps appear in the setup itself. This is the sort of targeted result that specialists in automorphism groups or linear groups over fields will find usable for follow-up questions. A reader already working on similar local-global problems gets immediate value from the concrete verification. It is not reshaping the field, but the evidence presented is checkable and grounded in known facts, so the central claims hold up. I would send it to a referee for a careful read on the details of the remaining cases.

Referee Report

0 major / 3 minor

Summary. The paper defines a local automorphism of a group G as a self-map f such that for every pair of elements x, y there exists an automorphism σ of G agreeing with f at x and y. It studies this notion for selected classical groups (linear, symplectic, and orthogonal groups over fields), proving that local automorphisms coincide with global automorphisms in some families and are otherwise closely related to them via the known structure of Aut(G).

Significance. The results clarify the local-to-global behavior of automorphisms for fundamental classical groups, using only standard facts about their automorphism groups and no ad-hoc parameters. This adds concrete evidence to the broader program of understanding when local interpolation properties imply global ones, with potential applications to rigidity questions and representation theory.

minor comments (3)

[§2] §2, Definition 2.1: the interpolation condition is stated for unordered pairs; confirm whether the definition requires ordered pairs or is symmetric, and add a sentence clarifying the relation to the usual notion of 2-local automorphism in the literature.
[§3, §4] Theorem 3.4 and Theorem 4.2: the statements that local automorphisms are 'closely related' to automorphisms would be strengthened by an explicit description (e.g., composition with a fixed inner automorphism or field automorphism) rather than the current qualitative phrasing.
[§1] The paper assumes the base field is infinite or has characteristic not 2 in several places; add a uniform statement of the field hypotheses at the beginning of each main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on local automorphisms of classical groups and the significance in the broader context of local-to-global properties. The recommendation for minor revision is noted, but no specific major comments were provided in the report. We will address any minor editorial or presentational issues in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces the standard definition of a local automorphism (a map interpolable by a global automorphism on every pair of elements) and then proves, for specific classical groups, that local automorphisms coincide with or are closely related to the known automorphism groups. This is a conventional local-to-global argument in group theory that relies on the external, independently established structure of automorphism groups of classical groups (e.g., via Dieudonné's theorem and related results). No equations reduce by construction to their own inputs, no parameters are fitted and then relabeled as predictions, and any self-citations (if present) are not load-bearing for the central claims; the results are externally falsifiable against the known automorphism groups. The derivation chain therefore contains no circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the newly introduced definition of local automorphism together with standard properties of classical groups and their automorphisms drawn from prior literature.

axioms (1)

standard math Standard axioms and properties of groups, automorphisms, and classical groups as established in the literature.
The paper invokes these without proof as background.

pith-pipeline@v0.9.0 · 5361 in / 972 out tokens · 84011 ms · 2026-05-07T17:27:47.034307+00:00 · methodology

discussion (0)

Reference graph

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