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arxiv: 1907.11479 · v1 · pith:D3IM3VMSnew · submitted 2019-07-26 · 🧮 math.GR

Lower central words in finite p-groups

Iker de las Heras , Marta Morigi This is my paper

Pith reviewed 2026-05-24 15:21 UTC · model grok-4.3

classification 🧮 math.GR
keywords lower central wordsp-groupsverbal subgroupsword valuespro-p groupsfinite groupscommutator words
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The pith

If γ_r(G) is 2-generated in a finite p-group G then every element of γ_r(G) is a γ_r-value, for every prime p and without any abelian assumption.

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

The paper proves that the verbal subgroup formed by a lower central word γ_r in a finite p-group consists entirely of single values of that word whenever the subgroup itself needs only two generators. This removes the earlier requirement that the subgroup be abelian and extends the statement from primes at least 5 down to p=3. The same conclusion is established for pro-p groups. The result matters because word values in groups do not in general form subgroups, so the 2-generator restriction supplies a concrete condition under which the verbal subgroup coincides exactly with its set of word values.

Core claim

For any prime p, if G is a finite p-group such that the verbal subgroup γ_r(G) is 2-generator, then γ_r(G) consists only of γ_r-values. The same holds when G is a pro-p group.

What carries the argument

The 2-generator condition on the verbal subgroup γ_r(G), which forces every element to be expressible as a single value of the lower central word rather than a product of several.

If this is right

  • The conclusion holds for p=3 as well as all larger primes.
  • The abelian hypothesis on γ_r(G) is unnecessary.
  • The identical statement is true for pro-p groups.
  • The set of γ_r-values equals the whole verbal subgroup under the stated generator bound.

Where Pith is reading between the lines

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

  • Similar generator restrictions might force other verbal subgroups to coincide with their word values in p-groups.
  • The result supplies a uniform description of the lower central series factors in pro-p groups that satisfy the 2-generator condition at each step.
  • Computational checks in small-order p-groups could verify the boundary between 2-generated and 3-generated cases.

Load-bearing premise

That the verbal subgroup γ_r(G) can be generated by only two elements.

What would settle it

A concrete finite p-group G in which γ_r(G) is 2-generated yet contains at least one element that is not equal to any single γ_r-value.

read the original abstract

It is well known that the set of values of a lower central word in a group $G$ need not be a subgroup. For a fixed lower central word $\gamma_r$ and for $p\ge 5$, Guralnick showed that if $G$ is a finite $p$-group such that the verbal subgroup $\gamma_r(G)$ is abelian and 2-generator, then $\gamma_r(G)$ consists only of $\gamma_r$-values. In this paper we extend this result, showing that the assumption that $\gamma_r(G)$ is abelian can be dropped. Moreover, we show that the result remains true even if $p=3$. Finally, we prove that the analogous result for pro-$p$ groups is true.

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 / 2 minor

Summary. The paper extends Guralnick's theorem on lower central words: for a finite p-group G with γ_r(G) 2-generated, it shows that γ_r(G) equals the set of γ_r-values without assuming γ_r(G) is abelian (for p ≥ 5 originally, now including p=3). It also proves the analogous statement for pro-p groups.

Significance. If correct, the result removes the abelian hypothesis from the prior theorem while retaining the explicit 2-generator condition, extends the statement to p=3, and adds the pro-p case. This clarifies the structure of verbal subgroups generated by lower central words in p-groups and pro-p groups. The work is presented as a direct extension with no hidden parameters or circular reductions.

minor comments (2)
  1. The abstract states the extensions but does not explicitly repeat the 2-generator hypothesis for the new claims (though it is retained from the Guralnick setup being extended).
  2. A brief remark on whether the proofs adapt existing techniques or require new arguments would help readers assess the technical novelty.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The report contains no specific major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity; derivation extends external theorem independently

full rationale

The paper's central claims extend Guralnick's prior theorem on verbal subgroups in p-groups by removing the abelian hypothesis, lowering the p threshold to 3, and adding the pro-p case, while explicitly retaining the 2-generator condition from the external setup. No load-bearing steps reduce by the paper's own equations to self-defined quantities, fitted inputs renamed as predictions, or self-citation chains; the cited result is independent (different authors) and the proofs are presented as direct extensions without internal redefinition or smuggling of ansatzes. This is the normal case of a self-contained extension of an external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard axioms of group theory and the definition of lower central series; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)
  • standard math Standard axioms of group theory (associativity, identity, inverses) and the definition of the lower central series γ_r(G).
    Invoked implicitly throughout any work on lower central words and verbal subgroups.

pith-pipeline@v0.9.0 · 5649 in / 1233 out tokens · 19279 ms · 2026-05-24T15:21:58.485869+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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