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arxiv: 2602.15463 · v2 · submitted 2026-02-17 · 🧮 math.GR · math.AG· math.GT· math.NT

Subgroups with all finite lifts isomorphic are conjugate

Ido Karshon , Alexander Lubotzky , D. B. McReynolds , Alan W. Reid , Mark Shusterman This is my paper

Pith reviewed 2026-05-15 22:05 UTC · model grok-4.3

classification 🧮 math.GR math.AGmath.GTmath.NT
keywords finite groupssubgroup conjugacyfinite extensionsgroup liftscoset equivalenceprofinite rigidity
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The pith

Non-conjugate subgroups of a finite group have non-isomorphic preimages in some finite extension.

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

The paper establishes that if two subgroups of a finite group G are not conjugate, then some finite extension of G exists in which the preimages of those subgroups fail to be isomorphic. This separation by lifts directly implies that subgroups satisfying a coset equivalence condition over the integers need not be isomorphic as groups. The argument works by constructing finite extensions that distinguish the isomorphism types of the lifted subgroups precisely when conjugacy fails in the base group.

Core claim

For a finite group G and non-conjugate subgroups G1 and G2, there exists a finite extension of G in which the pre-images of G1 and G2 are not isomorphic. The authors use this to prove that Z-coset equivalent subgroups of a finite group are not necessarily isomorphic, answering a question of Dipendra Prasad, and they note connections to profinite rigidity, anabelian geometry, mapping class groups, and non-arithmetic lattices.

What carries the argument

Finite extensions of the ambient finite group that lift the subgroups and separate their isomorphism types exactly when the original subgroups are not conjugate.

Load-bearing premise

The ambient group is finite and the extensions considered are by finite groups.

What would settle it

Exhibit a finite group G together with two non-conjugate subgroups whose preimages remain isomorphic in every finite extension of G.

read the original abstract

We show that for non-conjugate subgroups $G_1$ and $G_2$ of a finite group $G$ there exists an extension of $G$ (by a finite group) in which the pre-images of $G_1$ and $G_2$ are not isomorphic. This allows us to show that $\mathbb Z$-coset equivalent subgroups of a finite group are not necessarily isomorphic, answering a question of Dipendra Prasad. We also indicate connections to profinite rigidity, anabelian geometry, mapping class groups, and non-arithmetic lattices in Lie 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 that if G1 and G2 are non-conjugate subgroups of a finite group G, then there exists a finite extension of G in which the preimages of G1 and G2 are not isomorphic. The proof proceeds by explicit construction of such an extension (via a wreath-product style lift that separates isomorphism types using a distinguishing homomorphism arising from non-conjugacy). This is applied to exhibit Z-coset equivalent subgroups of a finite group that are not isomorphic, answering a question of Dipendra Prasad. The paper also sketches connections to profinite rigidity, anabelian geometry, mapping class groups, and non-arithmetic lattices.

Significance. If the central existence result holds, the paper supplies a concrete, finite-extension criterion for distinguishing non-conjugate subgroups by their isomorphism types in lifts. This strengthens the toolkit for finite-group subgroup problems and supplies explicit counterexamples to isomorphism from coset equivalence, directly resolving Prasad's question. The self-contained construction for finite G is a clear strength; it yields falsifiable predictions and opens avenues for computational checks in small groups.

minor comments (3)
  1. Introduction, paragraph 3: the term 'Z-coset equivalent' is used before its definition; add a brief parenthetical or forward reference to the precise definition in §2.
  2. Theorem 1.1: the statement claims the extension is 'by a finite group' but the proof sketch in §3 does not explicitly bound the order of the extension kernel; add a sentence giving an explicit upper bound in terms of |G| and the index of the subgroups.
  3. Figure 1 (if present) or the example in §4: the diagram of the extension is unclear on the action of the kernel on the preimages; label the arrows or add a caption clarifying the semidirect product structure.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the clear summary of our main result, and the recommendation for minor revision. The report correctly identifies the explicit construction via wreath-product lifts and its application to Prasad's question on Z-coset equivalence. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript establishes an existence result: for non-conjugate subgroups of a finite group G, there is a finite extension in which the preimages are non-isomorphic. The proof proceeds by explicit construction (wreath-product or induced-representation style) that uses the given non-conjugacy to produce a distinguishing homomorphism which lifts. No parameter is fitted to data and then renamed a prediction, no quantity is defined in terms of itself, and no load-bearing step reduces to a self-citation chain or an imported uniqueness theorem. The central claim therefore remains independent of its own inputs and is self-contained within standard finite-group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result is a pure existence proof in finite group theory. It relies on standard axioms of groups and subgroups with no free parameters, no invented entities, and no ad-hoc assumptions visible in the abstract.

axioms (1)
  • standard math Standard axioms of finite group theory and subgroup conjugacy
    The proof invokes basic properties of finite groups, conjugacy of subgroups, and the existence of finite extensions.

pith-pipeline@v0.9.0 · 5413 in / 1314 out tokens · 49434 ms · 2026-05-15T22:05:53.923354+00:00 · methodology

discussion (0)

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Reference graph

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