Normal Subgroups of Powerful $p$ -groups

James Williams

REVIEW 1 major objections 5 minor 9 references

Normal Subgroups of Powerful $p$ -groups

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Normal subgroups inside the p-th power subgroup of a powerful p-group are themselves powerfully nilpotent.

desk verdict Terse but correct note closing the expected gap in powerful p-groups; the skeleton is sound and the only real risk is the external order-bound theorem it quotes. read the letter →

arxiv 1908.07030 v1 pith:V44D2ET5 submitted 2019-08-19 math.GR

James Williams This is my paper · ORCID

classification math.GR MSC 20D15

keywords powerfulp-groupspowerfullynilpotentgroupsnormalsubgroupspowercommutatororderboundsfinitecoclass

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that for odd primes, every normal subgroup N of a powerful p-group G with $N \leq G^p$ is powerfully nilpotent; for $p=2$ the same conclusion holds when $N \leq G^4$. Powerfully nilpotent groups are powerful groups with a central series whose new steps satisfy $[H_i,G] \leq H_{i-1}^p$, and their rank and exponent are bounded in terms of a single invariant, the powerful coclass. The theorem upgrades the known fact that such $N$ is powerful to the full layered structure, and its corollary places powerfully nilpotent subgroups at every level of the lower $p$-power series of $G$. This also gives a partial answer to the question of which $p$-groups can appear as subgroups of powerful $p$-groups.

What carries the argument

The load-bearing mechanism is the commutator order bound stated as Theorem 3: in a powerful $p$-group, if $o(x) \leq p^{i+1}$ and $o(y) \leq p^i$, then $o([x^{p^j},y^{p^k}]) \leq p^{i-j-k}$. This lets the proof convert normality and containment in $G^p$ into precise order control on commutators, forcing elements of order $p^2$ to appear centrally in $N$ when $N$ has exponent $p^2$ (Lemmas 9--11). The induction closes with two quotient criteria: a finite $p$-group is powerfully nilpotent if and only if $G/G^{p^2}$ is, and if and only if $G/Z(G)^p$ is.

What would settle it

A direct counterexample would be a powerful $p$-group with odd $p$ and a normal subgroup $N \leq G^p$ such that $N/N^{p^2}$ is not powerfully nilpotent, since Proposition 6 then rules out powerful nilpotence of $N$. Concretely, an exhaustive check of powerful groups of order $3^k$ for $k \leq 8$, testing every normal subgroup contained in $G^3$, would either confirm the theorem or exhibit such $N$.

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Extended reading notes

Core claim

Theorem 12 states that if $p$ is odd, $G$ is powerful, and $N$ is normal in $G$ with $N \leq G^p$, then $N$ is powerfully nilpotent. Theorem 8 covers $p=2$ with $N \leq G^4$. The proof inducts on the order of $G$: first quotient by $N^{p^2}$, which reduces to the case where $N$ has exponent exactly $p^2$; a sequence of lemmas then finds an element of order $p^2$ in the centre of $N$, and quotienting by $Z(N)^p$ gives a smaller powerful quotient to which the inductive hypothesis applies. Proposition 6 and Proposition 7 translate powerful nilpotence of these quotients back to $N$, and the corollary extends the same conclusion to normal subgroups of $G^{p^i}$ contained in $G^{p^{i+1}}$ (with the case $p=2$ shifted by one).

Load-bearing premise

The argument stands on the external order bound stated as Theorem 3, which limits the order of every commutator of powers in a powerful $p$-group; if that bound fails in a single powerful $p$-group, the central-element lemmas that drive the induction collapse.

Editorial extensions

If this is right

Every normal subgroup of a powerful $p$-group that lies inside $G^p$ (or $G^4$ when $p=2$) is not only powerful but powerfully nilpotent, so it carries the full layered central-series structure.
By Corollary 13, normal subgroups of $G^{p^i}$ contained in $G^{p^{i+1}}$ (with the analogous shift for $p=2$) are powerfully nilpotent, making powerful nilpotence abundant in the lower $p$-power series.
Because rank and exponent of a powerfully nilpotent group are bounded by functions of its powerful coclass, the theorem brings quantitative control to deep normal subgroups of powerful $p$-groups.
The result partially answers the question of which $p$-groups embed in powerful $p$-groups: those that arise as normal subgroups inside $G^p$ have the strongly restricted structure of powerful nilpotence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof suggests that the only obstruction to powerful nilpotence in a powerful $p$-group lives outside the first power subgroup; normal subgroups that reach into $G^p$ lose the freedom to be wild, so the boundary case $N \not\leq G^p$ is where any counterexamples would have to hide.
One testable extension is whether the induction still works after replacing $G^p$ by lower terms $G^{p^k}$, which would broaden the corollary into a fuller characterization of which normal subgroups of a powerful $p$-group are powerfully nilpotent.
For $p=2$, the shift from $G^2$ to $G^4$ points to the quotient $N/N^4$ as the complete obstruction; computing the upper powerfully central series of that quotient may yield a criterion for powerful nilpotence of arbitrary normal subgroups in powerful $2$-groups.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a competent, useful note. It proves that for odd p, a normal subgroup N of a powerful p-group G with N ≤ G^p is powerfully nilpotent (and N ≤ G^4 for p=2). That's the natural strengthening of the González-Sánchez–Jaikin-Zapirain result that such N is powerful, and it partially answers Mann's question. The method is established powerful p-group technology; no new framework, but the commutator bookkeeping is careful and the argument holds.

Credit where due: Lemmas 9 through 11 and Theorem 12 are internally coherent. I checked the calls to Theorem 3 and they line up: Lemma 9 uses i=2, j=k=1; Lemma 10 uses i=3, j=2, k=1; Lemma 11 uses i=3, j=2, k=0 and i=3, j=3, k=0. The class-at-most-2 argument inside Lemma 11 is terse but legitimate, including the identity [b^p,a^p] = [b,a^p]^p [b,a^p,b]^{p choose 2}; p odd kills the second term because [[b,a^p],b] has order at most p. The induction in Theorem 12 is sound: first quotient by N^{p^2} if it's nontrivial, then quotient by Z(N)^p, which Lemma 11 guarantees is nontrivial, and apply Proposition 7.

Soft spots: the whole proof leans on Theorem 3 (Fernández-Alcober) at four points. That order bound is external and unproved here, and if the quoted version has a missing hypothesis, the argument collapses. That's a normal dependency in this area, but it means a referee can't verify the paper's core without trusting an outside theorem. The citations to the author's own [8] are also load-bearing—the quotient criteria in Propositions 6 and 7 and the fact that powerful 2-groups are powerfully nilpotent—but those are published in Journal of Algebra, so this is acceptable rather than suspect.

Minor: Proposition 7's proof is a one-liner that just restates the definition; it is correct, just terse. The p=2 case is handled separately and is fine. The corollary is immediate but harmless.

Who this is for: people working on finite p-group structure, particularly rank and coclass questions. It's not a landmark, but it closes a natural gap and is written clearly enough to skim in ten minutes.

Recommendation: send it to peer review. A competent referee can check the commutator steps without much pain, and the result is a worthwhile short note for the literature.

Referee Report

1 major / 5 minor

Summary. The paper proves that if G is a finite powerful p-group and N is a normal subgroup of G contained in G^p (for odd p) or in G^4 (for p=2), then N is powerfully nilpotent. The proof reduces the problem to the case where N has exponent p^2, uses commutator-order estimates from Fernández-Alcober's Theorem 3 to show that N then contains a nontrivial central element of order p^2 (Lemma 11), and finishes by induction on |G| together with two reduction criteria (Propositions 6 and 7). The p=2 case is handled by proving that N/N^4 is abelian, so that N is a powerful 2-group and hence powerfully nilpotent by a result from [8].

Significance. The main theorem is a natural strengthening of the earlier result of González-Sánchez and Jaikin-Zapirain, which showed only that such a subgroup N is powerful. It also gives a partial answer to Mann's question about which p-groups occur as subgroups of powerful p-groups. The proof is concise, and the commutator estimates are applied carefully; the role of the external Theorem 3 is explicitly identified. If the cited results are correct, the central claim follows from the presented argument. The paper is a well-focused research note rather than a survey or a computational study.

major comments (1)

[Section 2, Proposition 7] The proof of Proposition 7 is a single sentence, but this reduction is load-bearing: it is used in the induction step of Theorem 12 to conclude that N is powerfully nilpotent from the powerful nilpotence of N/Z(N)^p. Please provide a full proof, or a precise citation to [8]. The nontrivial point is that when one lifts the upper powerfully central series from G/Z(G)^p to G, one must verify that the lifted subgroups H_i satisfy [H_i,G] ≤ H_{i-1}^p at every step; this requires an inductive argument using Z(G)^p ⊆ H_{i-1}^p, and it is not immediate from the definition alone.

minor comments (5)

[Theorem 3] Please give the exact lemma or proposition number in [2] where Theorem 3 is proved, since Lemmas 9, 10, and 11 and Theorem 8 all rely on this bound.
[Lemma 9] In the proof of Lemma 9, the application of Theorem 3 uses j=k=1; please state this explicitly so the reader can verify the exponent subtraction i-j-k = 2-1-1 = 0.
[Theorem 12] After the sentence 'for otherwise N would be abelian by Lemma 9', please add explicitly that an abelian p-group is powerfully nilpotent, so the argument may indeed assume that the exponent of N is exactly p^2.
[Corollary 13] The corollary is labelled 'immediate' but would be clearer if it noted that G^{p^i} is itself a powerful p-group and that Theorem 12 (or Theorem 8 for p=2) is applied with H = G^{p^i}.
[Throughout] The text contains several spacing and typographical artifacts such as 'o f', 'powerf ul', and 'subg roup'; these should be corrected in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 12 is proved from the external commutator bound of Fernández-Alcober via induction; self-citations are tool citations only.

full rationale

The derivation of Theorem 12 is not circular. Lemmas 9, 10, and 11 each reduce a commutator of pth powers directly to the external Theorem 3, a commutator-order bound proved by Fernández-Alcober, not by the present author. For example, Lemma 10 applies Theorem 3 with i=3, j=2, k=1 to obtain [g^{p^2},h^p]=1, and Lemma 11 uses the same bound to force [[[b,a^p],b],b]=1. The induction in Theorem 12 then uses two quotient criteria from the author's prior work: Proposition 6 (powerfully nilpotent iff the p^2-power quotient is powerfully nilpotent) and Proposition 7 (powerfully nilpotent iff the quotient by Z(G)^p is powerfully nilpotent). These are general structural criteria for the class of powerfully nilpotent groups, not restatements of the theorem being proved, and Proposition 7 is even given a one-line proof from the definition. The p=2 case invokes the prior result that powerful 2-groups are powerfully nilpotent from [8], but this is a published external classification, and it is not used in the odd-prime main proof. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported to force a choice, and no definition is circularly tied to the target conclusion. The central claim therefore has independent mathematical content derived from an external commutator bound and induction.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The proof is a standard group-theoretic derivation. It introduces no free parameters and no invented entities. It does rely on a package of established theorems about powerful p-groups, several from the author's earlier work, which are treated as black boxes.

assumptions (9)

domain assumption All groups considered are finite p-groups.
Section 2, first sentence; the theorem is stated only in this category, and the proof uses finiteness for induction on order.
standard math For a powerful p-group, G^{p^k} is exactly the set of p^k-th powers, is powerfully embedded, and G^p is generated by p-th powers of generators.
Theorem 1 from [5, Lubotzky-Mann], used throughout to write elements of G^p as p-th powers.
standard math Powerfully embedded subgroups satisfy [M^{p^i}, N^{p^j}] = [M,N]^{p^{i+j}}.
Lemma 2 from [7, Shalev], used to justify inclusions such as [G, G^p] ≤ G^{p^2}.
standard math Fernandez-Alcober's bound: if o(x) ≤ p^{i+1} and o(y) ≤ p^i, then o([x^{p^j}, y^{p^k}]) ≤ p^{i-j-k}.
Theorem 3 is the engine of Lemmas 9, 10, 11 and the p=2 case in Theorem 8.
standard math A normal subgroup N ≤ G^p of a powerful p-group is powerful, with the analogous statement N ≤ G^4 for p=2.
Theorem 4 from [3, Gonzalez-Sanchez and Jaikin-Zapirain], cited in the introduction as the starting point.
standard math G is powerfully nilpotent if and only if G/G^{p^2} is powerfully nilpotent.
Proposition 6 from [8], used in the induction step of Theorem 12.
standard math G is powerfully nilpotent if and only if G/Z(G)^p is powerfully nilpotent.
Proposition 7 from [8], used at the end of Theorem 12.
standard math Every powerful 2-group is powerfully nilpotent.
Result from [8, page 81], used to finish the p=2 case in Theorem 8.
standard math For a class-2 group, [x^n, y] = [x,y]^n [x,y,x]^{n choose 2}.
Used in Lemma 11 to compare [b^p, a^p] with [b,a^p]^p.

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Cite this review

Pith. "Pith review of Normal Subgroups of Powerful $p$ -groups." pith.science (2026). https://pith.science/paper/V44D2ET5

@misc{pith2026190807030,
  author       = {Pith},
  title        = {Pith review of: Normal Subgroups of Powerful $p$ -groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V44D2ET5}},
  note         = {Machine review of arXiv:1908.07030}
}

abstract

In this note we show that if $p$ is an odd prime and $G$ is a powerful $p$-group with $N\leq G^{p}$ and $N$ normal in $G$, then $N$ is powerfully nilpotent. An analogous result is proved for $p=2$ when $N\leq G^{4}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[8]

Gunnar Traustason and James Williams, Powerfully nilpotent groups , Journal of Algebra 522 (2019), 80 – 100

work page 2019
[1]

Dixon, M.P.F

J.D. Dixon, M.P.F. Du Sautoy, A. Mann, and D. Segal, Analytic pro-p groups , Cambridge Studies in Advanced Mathematics, Cambridge University Pre ss, 2003

work page 2003
[2]

Fernández-Alcober, Omega subgroups of powerful p-groups , Israel Journal of Math- ematics 162 (2007), no

Gustavo A. Fernández-Alcober, Omega subgroups of powerful p-groups , Israel Journal of Math- ematics 162 (2007), no. 1, 75–79

work page 2007
[3]

González-Sánchez and A

J. González-Sánchez and A. Jaikin-Zapirain, On the structure of normal subgroups of potent p-groups, Journal of Algebra 276 (2004), no. 1, 193 – 209

work page 2004
[4]

Khukhro, P-automorphisms of ﬁnite p-groups , Lecture note series / London mathematical society, Cambridge University Press, 1998

E.I. Khukhro, P-automorphisms of ﬁnite p-groups , Lecture note series / London mathematical society, Cambridge University Press, 1998

work page 1998
[5]

Alexander Lubotzky and A vinoam Mann, Powerful p-groups. i. ﬁnite groups , Journal of Algebra 105 (1987), no. 2, 484 – 505

work page 1987
[6]

A vinoam Mann, Some questions about p-groups , Journal of the Australian Mathematical Soci- ety 67 (1999), 356 – 379

work page 1999
[7]

Shalev, On almost ﬁxed point free automorphisms , Journal of Algebra 157 (1993), no

A. Shalev, On almost ﬁxed point free automorphisms , Journal of Algebra 157 (1993), no. 1, 271 – 282

work page 1993

Show all 9 references

[9]

DOI: 10.22108/ijgt.2019.113217.1507

James Williams, Omegas of agemos in powerful groups , International Journal of Group Theory, –, In press. DOI: 10.22108/ijgt.2019.113217.1507

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [8]

Gunnar Traustason and James Williams, Powerfully nilpotent groups , Journal of Algebra 522 (2019), 80 – 100

work page 2019

[2] [1]

Dixon, M.P.F

J.D. Dixon, M.P.F. Du Sautoy, A. Mann, and D. Segal, Analytic pro-p groups , Cambridge Studies in Advanced Mathematics, Cambridge University Pre ss, 2003

work page 2003

[3] [2]

Fernández-Alcober, Omega subgroups of powerful p-groups , Israel Journal of Math- ematics 162 (2007), no

Gustavo A. Fernández-Alcober, Omega subgroups of powerful p-groups , Israel Journal of Math- ematics 162 (2007), no. 1, 75–79

work page 2007

[4] [3]

González-Sánchez and A

J. González-Sánchez and A. Jaikin-Zapirain, On the structure of normal subgroups of potent p-groups, Journal of Algebra 276 (2004), no. 1, 193 – 209

work page 2004

[5] [4]

Khukhro, P-automorphisms of ﬁnite p-groups , Lecture note series / London mathematical society, Cambridge University Press, 1998

E.I. Khukhro, P-automorphisms of ﬁnite p-groups , Lecture note series / London mathematical society, Cambridge University Press, 1998

work page 1998

[6] [5]

Alexander Lubotzky and A vinoam Mann, Powerful p-groups. i. ﬁnite groups , Journal of Algebra 105 (1987), no. 2, 484 – 505

work page 1987

[7] [6]

A vinoam Mann, Some questions about p-groups , Journal of the Australian Mathematical Soci- ety 67 (1999), 356 – 379

work page 1999

[8] [7]

Shalev, On almost ﬁxed point free automorphisms , Journal of Algebra 157 (1993), no

A. Shalev, On almost ﬁxed point free automorphisms , Journal of Algebra 157 (1993), no. 1, 271 – 282

work page 1993