On non self-normalizing subgroups

Emanuele Pacifici; Lucia Sanus; Mariagrazia Bianchi; Mark L. Lewis; Rachel D. Camina

REVIEW 2 major objections 4 minor 14 references

On non self-normalizing subgroups

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Counting conjugacy classes of non-self-normalizing subgroups yields a complete description of $D_0$ through $D_4$, with the only nonsolvable group in $D_4$ being $A_5$.

desk verdict Genuinely new small-n classification of a subgroup-counting invariant, with a likely-correct D4 dichotomy that needs reproducible computational checks. read the letter →

arxiv 2411.18102 v1 pith:UCWXBHZJ submitted 2024-11-27 math.GR

Mariagrazia Bianchi , Rachel D. Camina , Mark L. Lewis , Emanuele Pacifici , Lucia Sanus This is my paper · ORCID

classification math.GR MSC 20E0720E3420D1020D15

keywords nonself-normalizingsubgroupsconjugacyclassessolvablegroupsderivedlengthnilpotencyclassFrobeniusfinitesimpleD4classification

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

Finite groups can be sorted by a single number: $D(G)$, the number of conjugacy classes of nontrivial subgroups that are not self-normalizing. The paper studies the families $D_n$ of groups with $D(G)=n$ and asks how this one number constrains solvability, derived length, and nilpotency class. Its main result is a complete dichotomy for $n=4$: a nonsolvable group in $D_4$ must be the alternating group $A_5$, and every other group in $D_4$ is solvable, with derived length at most $2$ unless it is isomorphic to $SL_2(3)$. Along the way it classifies $D_0$ through $D_3$, proves those groups are solvable of derived length at most $2$, bounds nilpotency class and derived length for nilpotent groups in $D_n$, and obtains a general derived-length bound for solvable groups in $D_n$. If the paper is right, the single invariant $D(G)$ determines group structure much more tightly than the mere absence of self-normalizing subgroups would suggest.

What carries the argument

The engine of the paper is the invariant $D(G)$, the number of $G$-conjugacy classes of nontrivial subgroups $H$ with $N_G(H)\neq H$. The load-bearing tool is the quotient lemma (Lemma 3.2): if $N$ is a nontrivial proper normal subgroup, then $D(G/N)\le D(G)-1$, because every non-self-normalizing subgroup class of the quotient lifts to one of $G$ not properly contained in $N$. This lets the authors descend from a group to its quotients, reducing questions about $D_n$ to small cases. For the $D_4$ dichotomy, the final step combines this descent with the finite list of simple groups whose order is divisible by exactly three primes, and with a computer search over groups of order $48$. The Frobenius-group counts in Section 5 provide the explicit classifications of $D_2$ and $D_3$.

What would settle it

Re-run the two finite searches that Theorem 8.2 and Theorem 8.3 depend on: compute $D(G)$ for every simple group whose order has exactly three prime divisors, and enumerate all groups of order 48 to see whether any has a nonabelian Sylow 2-subgroup coinciding with its derived subgroup. A single counterexample in either search would overturn the claimed uniqueness.

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Extended reading notes

Core claim

Define $D(G)$ as the number of conjugacy classes of nontrivial subgroups $H$ of a finite group $G$ with $N_G(H)\neq H$; $D_n$ is the family of groups with $D(G)=n$. The central claim is that $D_4$ is completely understood: by Theorem 8.2, if $G$ is nonsolvable and $G\in D_4$, then $G\cong A_5$; by Theorem 8.3, every other group in $D_4$ is solvable and either has derived length at most $2$ or is isomorphic to $SL_2(3)$. The same invariant yields explicit classifications for $n\le 3$: all groups in $D_0,D_1,D_2,D_3$ are solvable with derived length at most $2$, with the exceptional examples listed as cyclic, Frobenius, and $A_4$ groups. Nilpotent groups in $D_n$ have nilpotency class at most $n/2$ and derived length at most $\log_2(n/2)+1$, and solvable groups in $D_n$ have derived length at most $\min(n-1,3\log_2(n+1)+9)$. The paper also proves a product inequality $D(H\times K)\ge (D(H)+2)(D(K)+2)-2$, with equality exactly when $H$ and $K$ are nilpotent of coprime orders.

Load-bearing premise

The load-bearing premise is that the finite computational assertions in Section 8 are correct: the paper reports a 'direct check' giving $A_6\in D_{11}$, $L_2(7)\in D_8$, $L_2(8)\in D_6$ while leaving $A_5$ as the only simple candidate, and asserts that a computational search finds no group of order 48 with a nonabelian Sylow 2-subgroup equal to its derived subgroup. If either of these unscripted checks is wrong, the uniqueness of $A_5$ and $SL_2(3)$ in $D_4$ would need re-examination.

Editorial extensions

If this is right

Every group with $D(G)\le 3$ is solvable of derived length at most $2$, so small values of the invariant force solvability.
The only nonsolvable group with $D(G)=4$ is $A_5$; no other simple or almost-simple group has exactly four conjugacy classes of non-self-normalizing subgroups.
Every solvable group in $D_4$ has derived length at most $3$, and the only one reaching $3$ is $SL_2(3)$.
Nilpotent groups in $D_n$ have nilpotency class at most $n/2$ and derived length at most $\log_2(n/2)+1$, so the count of non-self-normalizing subgroup classes bounds how deep the group is.
For solvable groups, membership in $D_n$ bounds derived length by $\min(n-1,3\log_2(n+1)+9)$, a constraint that grows only logarithmically in $n$.

Reading between the lines

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

The $D_4$ dichotomy makes it plausible that $D_5$ contains only solvable groups; enumerating the finite groups of the relevant orders would test this directly.
The product inequality suggests that direct products of nilpotent groups of coprime orders realize many $D$-values exactly, so one could ask whether every sufficiently large integer occurs as $D(G)$ for some finite group.
The uniqueness of $A_5$ and $SL_2(3)$ currently rests on unscripted computer checks described in Section 8; formalizing those two finite searches would make the classification machine-checkable, and any error there would reopen the theorem.
The paper's own computational experiments indicate the bound $3\log_2(n+1)+9$ is far from sharp; improving it, or showing that all $D_5$ groups have derived length at most $3$, would be the next quantitative step.

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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

You should know two things about this paper. First, it is a real piece of finite group theory: it introduces a new counting invariant D(G), classifies D0 through D3 completely, proves that A5 is the only nonsolvable group in D4, and shows SL2(3) is the only solvable D4-group of derived length 3. Second, the D4 dichotomy is the headline result, and it is probably right, but it rests on a couple of finite computational assertions that are stated without code or transcripts.

What the paper does well: the counting arguments are coherent and mostly checkable by hand. Lemma 3.2 (the quotient bound) is a nice tool, the Frobenius counts in Section 5 are explicit and I spot-checked several of them, and the nilpotent bounds via the direct-product inequality are clean. The authors are also honest about the limits of their methods: they say the D2/D3 lists are already complicated, they flag the GAP check explicitly, and they do not oversell the logarithmic bound in Theorem 8.4, which they admit is much larger than what computations suggest. Citation practice looks fair; the Herzog list and Glasby's theorem are standard, and the self-citation to the related non-self-centralizing paper is appropriate.

Soft spots, in proportion: the main one is Section 8. In Theorem 8.2, after reducing to Herzog's eight simple groups, the proof says a direct check rules out all but A5 and reports D(A6)=11, D(L2(7))=8, D(L2(8))=6. Those values are load-bearing: they separate A5 from the rest. In Theorem 8.3, the final case |G|=48 uses a GAP assertion that no group of order 48 has a nonabelian Sylow 2-subgroup equal to the derived subgroup. Neither check comes with code, a transcript, or a counting formula. This is a reproducibility gap, not a flaw I can point to in the logic—I verified several of the smaller counts myself and found no error—but it should be fixed before the paper is final. Minor: Proposition 3.3 asserts S4 lies in D7 without proof, and the phrase direct check appears a couple of times where a short argument would be better. Both are easily patched.

Overall: the central argument holds up on reading. The classifications are new, the reductions are careful, and I do not see a logical hole. The paper is for finite group theorists who work on subgroup lattices, normalizers, or solvability criteria. It deserves a serious referee; the referee should ask for the computational evidence to be made available, ideally as a short appendix or supplementary file. I would send it to peer review with that condition.

Referee Report

2 major / 4 minor

Summary. The paper introduces and studies the family D_n of finite groups with exactly n conjugacy classes of nontrivial subgroups that are not self-normalizing. The main structural results are: groups in D_n for n ≤ 3 are solvable with derived length at most 2 (Proposition 3.3); A5 is the unique nonsolvable group in D4 (Theorem 8.2); SL2(3) is the unique solvable group in D4 with derived length greater than 2, all other solvable groups in D4 having derived length at most 2 (Theorem 8.3). The paper also proves a lower bound for D(H × K) with an equality criterion (Lemma 4.2), bounds on the nilpotency class and derived length of nilpotent groups in D_n (Theorems 4.1 and 4.3), a classification of D0–D3 (Propositions 3.1, 3.4, 7.1, 7.2), and a general bound on the derived length of solvable groups in D_n (Theorem 8.4). The methods are elementary group theory: Sylow theory, Burnside's normal p-complement theorem, Frobenius group structure, Maschke's theorem, and Z-group structure.

Significance. If the D4 classification is correct, it is a clean and complete answer for the first nonsolvable value of n, and the paper provides several reusable tools: the factor-group lemma (Lemma 3.2), the product inequality (Lemma 4.2), and the Frobenius-group counts (Section 5). The general derived-length bound (Theorem 8.4) and the nilpotency-class bounds are reasonable first steps toward a broader theory. The paper is well-written and the non-computational arguments are coherent, relying on standard theorems. The manuscript also honestly states its computational steps and points to related work by Jones. The main weakness is that the two decisive finite computations in Section 8 are asserted without reproducible code or a complete hand-checkable derivation, which means the central D4 dichotomy is not fully demonstrated as written.

major comments (2)

[Section 8, Theorem 8.2] The proof eliminates all simple groups in Herzog's list except A5 by asserting that 'a direct check rules out all the others except A5' and reporting D(A6)=11, D(L2(7))=8, D(L2(8))=6, while the last four groups are dismissed 'because of the orders of their Sylow subgroups.' These assertions are load-bearing: without them the theorem only shows that a nonsolvable group in D4 must be almost simple with socle on this list. Since neither the D-values nor the Sylow-order argument are proved or documented, this step is not reproducible. Please provide the GAP/Magma code and output, or a hand-checkable counting argument, for each of the eight groups in the list, including the four groups excluded by the orders of their Sylow subgroups.
[Section 8, Theorem 8.3] The proof uses the assertion 'it can be checked via GAP that there do not exist groups of order 48 with a nonabelian Sylow 2-subgroup coinciding with the derived subgroup' to conclude |N|=2 and hence G≅SL2(3). This is a nontrivial finite enumeration over all groups of order 48 and is essential to the uniqueness of SL2(3) in D4 with derived length 3. The paper does not include the GAP code, the exact search predicate, or the output. Please provide these, or give a mathematical proof that no such group exists.

minor comments (4)

[Theorem 4.3 proof] In the sentence 'the nilpotency class of D(P_j) is 1', the symbol D(P_j) appears to be a typo for P_j; the statement should concern the nilpotency class of the Sylow subgroup P_j.
[Theorem 8.3 proof] The reference 'By Proposition 8.2, it follows that G is solvable' should be 'By Theorem 8.2', since Proposition 8.2 does not exist.
[Proposition 7.2, case (6)] In the proof of the forward direction for the Frobenius group C_pr ⋊ C_q, the text says G has three normal subgroups 'of orders p, q, pr respectively'; the subgroup of order q is the Frobenius complement and is not normal in a nontrivial Frobenius group, so this should read 'of orders p, r, pr respectively'.
[Section 5, Lemmas 5.2 and 5.3] The displayed formulas for D(G) in the statements of Lemmas 5.2 and 5.3 have ambiguous fraction formatting; for example, 'D p+1 q +1' should be typeset as D_{(p+1)/q+1} to avoid confusion between (p+1)/q+1 and (p+1)/(q+1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivations are direct counting arguments with external standard theorems and explicitly stated finite computations, none of which are fitted to or defined by the target results.

full rationale

The paper defines D_n and D(G) as counts of conjugacy classes of nontrivial non-self-normalizing subgroups, and the central results are obtained by direct subgroup-lattice counting, induction on group order, and standard external theorems (Burnside's normal p-complement theorem, Frobenius/Thompson structure, Herzog's finite list of simple groups with three prime divisors, and Glasby's derived-length bound). The D4 dichotomy in Theorems 8.2 and 8.3 rests on finite checks reported in the text, such as 'a direct check rules out all the others except A5: in fact we have A6 ∈ D11, L2(7) ∈ D8 and L2(8) ∈ D6' and 'it can be checked via GAP that there do not exist groups of order 48 with a nonabelian Sylow 2-subgroup coinciding with the derived subgroup.' These are explicit computational assertions about concrete groups, not parameters fitted to force A5 or SL2(3); if they were wrong the theorem would fail, but that is a reproducibility and verification concern, not circularity. No equation defines D_n in terms of A5 or SL2(3), no boundary case is eliminated by citing the conclusion, and no load-bearing step is justified by a self-citation whose content is the target result. The paper's use of citations, including the Glasby result and Herzog's list, is standard external support. Accordingly, the honest finding is no circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters appear: D(G) is defined by counting conjugacy classes, and all stated parameters n, k, p, q are variables in universal or existential claims. The axioms are standard external theorems in finite group theory; no entity is introduced ad hoc.

assumptions (6)

standard math Frobenius kernel is a normal nilpotent subgroup and Frobenius complements have cyclic or generalized quaternion Sylow subgroups.
Section 2 background; used in Section 5 to count non-self-normalizing subgroups of Frobenius groups and in the D2/D3 classifications.
standard math Burnside's normal p-complement theorem: if a Sylow p-subgroup is contained in the center of its normalizer, then the group has a normal p-complement.
Stated as Theorem 2.1; used in Proposition 3.3 and Theorem 8.2 to force contradictions from self-normalizing abelian Sylow subgroups.
standard math Z-group structure theorem: if all Sylow subgroups of G are cyclic, then G/G' and G' are cyclic of coprime orders.
Stated as Theorem 2.2 and used in Proposition 3.3 and Theorem 8.2 to reduce to solvable cases.
standard math Maschke's theorem and irreducibility of GF(p)[C]-modules.
Used in Lemmas 5.2 and 5.3 to count orbits of a Frobenius complement acting on subgroups of elementary abelian kernels.
standard math Herzog's list of simple groups whose order is divisible by exactly three primes: A5, A6, L2(7), L2(8), L2(17), L3(3), U3(3), U4(2).
Used in Theorem 8.2 to reduce the nonsolvable D4 analysis to eight candidate simple groups; this list is a consequence of the classification of finite simple groups.
standard math Glasby's theorem bounding derived length of a solvable group in terms of its composition length.
Used in Theorem 8.4 to convert the composition length bound n+1 into the derived length bound 3 log2(n+1)+9.

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Cite this review

Pith. "Pith review of On non self-normalizing subgroups." pith.science (2026). https://pith.science/paper/UCWXBHZJ

@misc{pith2026241118102,
  author       = {Pith},
  title        = {Pith review of: On non self-normalizing subgroups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UCWXBHZJ}},
  note         = {Machine review of arXiv:2411.18102}
}

abstract

Let $n$ be a non negative integer, and define $D_n$ to be the family of all finite groups having precisely $n$ conjugacy classes of nontrivial subgroups that are not self-normalizing. We are interested in studying the behavior of $D_n$ and its interplay with solvability and nilpotency. We first show that if $G$ belongs to $D_n$ with $n \le 3$, then $G$ is solvable of derived length at most 2. We also show that $A_5$ is the unique nonsolvable group in $D_4$, and that $SL_2(3)$ is the unique solvable group in $D_4$ whose derived length is larger than 2. For a group $G$, we define $D(G)$ to be the number of conjugacy classes of nontrivial subgroups that are not self-normalizing. We determine the relationship between $D(H \times K)$ and $D(H)$ and $D(K)$. We show that if $G$ is nilpotent and lies in $D_n$, then $G$ has nilpotency class at most $n/2$ and its derived length is at most $\log_2 (n/2) + 1$. We consider $D_n$ for several classes of Frobenius groups, and we use this classification to classify the groups in $D_0$, $D_1$, $D_2$, and $D_3$. Finally, we show that if $G$ is solvable and lies in $D_n$ with $n \ge 3$, then $G$ has derived length at most the minimum of $n-1$ and $3 \log_2 (n+1) + 9$.

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Bosma, J

W. Bosma, J. Cannon, C. Playout, The Magma algebra system. I. The user language , J. Symbolic Comput. 24 (1997), 235–265

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C. Delizia, U. Jezernik, P. Moravec, C. Nicotera, C. Park er, Locally ﬁnite groups in which every noncyclic subgroup is self-centralizing, Journal of Pure and Applied Algebra, 221(2), (2017), 401-4 10

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