REVIEW 5 minor 12 references
A Finite E-Group of Nilpotency Class Three
T0 review · 0 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read This paper proves that a finite 3-group of order $3^{84}$ is an E-group of nilpotency class three.
desk verdict Answers a named open problem with a clean reduction to an exhaustive finite check; the imported structural facts are the only real dependency. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the relation tensor $q:V\to\Lambda^2 V$ assembled from the nine cube relations of $P$, together with the notion of a $q$-closed subspace $U$, meaning $q(U)\subseteq\Lambda^2 U$. For a bivector, its support is the image of the contraction map $V^*\to V$, and Lemma 4.1 characterizes $\omega\in\Lambda^2 U$ by the condition that the support of $\omega$ lies in $U$. Iterating support enlargements from any nonzero vector gives the least $q$-closed subspace containing it, so it suffices to check one representative of each of the 9841 projective directions of $\mathrm{PG}(8,3)$. The finite certificate is an exact Gaussian-elimination computation over $\mathbb{F}_3$, reproduced in Appendix A.
What would settle it
Run the exact closure computation of Appendix A over all 9841 projective points of $\mathrm{PG}(8,3)$; if any nonzero proper subspace $U\leq V$ satisfies $q(U)\subseteq\Lambda^2 U$, or if the rank distribution and growth profiles do not reproduce, the rigidity theorem is false. Independently, recompute the structure of $P$ and check whether $\Omega_1(P')=Z(P)$; if that equality fails, the singular-endomorphism step collapses.
Extended reading notes
Core claim
The central claim, stated on the paper's terms, is Theorem 5.1: the group $P$ is a finite E-group of nilpotency class three. For any endomorphism $\varphi$ of $P$, the induced map $L$ on $V=P/\Phi(P)$ obeys the compatibility law $q\circ L=(\Lambda^2 L)\circ q$, so the image subspace $\operatorname{im} L$ is $q$-closed. The paper proves by exact exhaustive computation over the 9841 points of $\mathrm{PG}(8,3)$ that the only $q$-closed subspaces of $V$ are $0$ and $V$. If $\operatorname{im} L=V$, Burnside's basis theorem makes $\varphi$ an automorphism, and the known A-group property gives commutation. If $\operatorname{im} L=0$, the image lies inside $\Phi(P)=P'$; applying $\varphi$ to the nine cube relations kills all commutator terms, forcing each generator image to have order dividing three, and the identity $\Omega_1(P')=Z(P)$ then puts the entire image in the center. In both cases every element commutes with its endomorphic image.
Load-bearing premise
The load-bearing premise is the published structural dossier on $P$: order $3^{84}$, nilpotency class three, $\Phi(P)=P'$, $P'$ abelian, $\Omega_1(P')=Z(P)$, and the A-group property; if any one of these recorded facts is wrong, the E-group conclusion does not follow.
Editorial extensions
If this is right
- Problem 11.46(a) of the Kourovka Notebook has an affirmative answer: finite E-groups of nilpotency class three exist.
- Every endomorphism of $P$ acts on the Frattini quotient $V$ either invertibly or as the zero map, because only $0$ and $V$ are $q$-closed.
- When the induced map is zero, the endomorphic image is forced into $Z(P)$, so the class-three group has a mechanism in which trivial Frattini action implies central image.
- The dual anticommutative algebra of $q$ is simple, since $q$-closed subspaces correspond exactly to ideals.
- The proof's finite certificate is checkable from the paper alone through the exact verifier in Appendix A.
Reading between the lines
- My inference: the $q$-closed subspace test can be run on other class-three candidates; a group whose tensor admits a nonzero proper $q$-closed subspace might admit a singular endomorphism with noncentral image, so this criterion separates E-groups from near-E-groups.
- My inference: the trivial linear stabilizer of this tensor explains why no projective orbit compression is available for the certificate, and finding a rigid tensor with a larger stabilizer could yield shorter certificates or smaller examples.
- My inference: the paper leaves open whether every finite E-group of class three must satisfy the same invertible-or-zero dichotomy on the Frattini quotient, and that would be a natural structural question to test next.
- My inference: the equivalence between $q$-closed subspaces and ideals suggests a search strategy for further examples by classifying simple anticommutative algebras over $\mathbb{F}_3$ with a compatible power structure, rather than by enumerating groups directly.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that the 3-group P of order 3^84 studied by Abdollahi, Faghihi, Linton, and O'Brien is an E-group of nilpotency class three, giving a positive answer to Kourovka Notebook Problem 11.46(a). The argument passes to the Frattini quotient V = P/Φ(P) ≅ F_3^9, encodes the nine cube relations as a linear map q: V → Λ^2V, and proves the rigidity statement that the only subspaces U ≤ V with q(U) ⊆ Λ^2U are 0 and V. The proof of this rigidity is an exact finite enumeration over the 9841 points of PG(8,3), with an executable verifier printed in full in Appendix A. A naturality argument shows that the image of any endomorphism on V is q-closed, so every endomorphism acts on V either invertibly or trivially. The invertible case is handled by the previously established A-group property of P, and the zero case is forced into Z(P) using the cube relations and the published identity Ω_1(P') = Z(P).
Significance. If correct, this gives the first finite E-group of nilpotency class three and settles a listed open problem. The main conceptual contribution is the reduction of the endomorphism problem to a rigidity property of a single tensor over F_3, and the finite certificate is genuinely machine-checked: the verifier in Appendix A is printed in full, is independent of the E-group claim, and includes assertions for the projective count, the rank distribution, the closure dimensions, the growth profile, and a negative control. The paper also gives a clean dual description of the rigidity as simplicity of a nine-dimensional anticommutative algebra. The proof legitimately relies on published structural facts about P, which is a normal external premise; the new tensor rigidity and the class-three lifting argument are the actual contributions.
minor comments (5)
- [Section 2, Eq. (4) and following] The notation "lower exponent-3 central series" for P_2(P) is inconsistent with the quoted |P/P_2(P)| = 3^45 and with Φ(P) = P': under the standard definition P_2(P) = P'P^3 = P' would have index 3^9. The subsequent arguments require P_2(P) to be the fully invariant subgroup γ_3(P) = Z(P), the quotient by which has order 3^45. Please state this identification explicitly and justify full invariance from that definition.
- [Section 2, paragraph after |P/P_2(P)| = 3^45] The sentence "so |Φ(P)| = 3^{45-9} = 3^{36}" is missing an overline on Φ; it should read |Φ(\bar P)| = 3^{36}. As printed it contradicts |Φ(P)| = |P'| = 3^75 from (4).
- [Proposition 4.5] The proof invokes "Aut(P) = Aut_c(P)" from [3] for an automorphism of \bar P, but the cited result is stated for P itself, not for the class-two quotient \bar P. Since Remark 4.6 explicitly says this proposition is not needed for the E-group theorem, please either supply a transfer argument or weaken the claim to what is actually proved.
- [Equation (6)] The left-hand side "[\bar x_i,\bar x_j]P_2(P)" is written in a way that suggests a double quotient; it should denote the element [\bar x_i,\bar x_j] of \bar P_2? = \bar P', identified with e_i ∧ e_j.
- [Throughout] The structural data in Proposition 2.1 would be much easier to read with standard notation (C_9)^{36} × (C_3)^3 and (C_3)^{39}; the inline plain-text forms "C36 9 × C3 3" and "C39 3" are ambiguous.
Circularity Check
No circularity: the finite tensor rigidity check and the imported structural facts are independent of the E-group claim.
full rationale
The paper's derivation chain is not circular. The relation tensor q is fixed directly by the explicit nine relation rows of the presentation (3), not by the E-group conclusion. Lemma 3.1 derives the compatibility condition q(Lv) = (Lambda^2 L)q(v) from functoriality of cubing and commutators, which is a genuine one-way implication usable to constrain endomorphisms. Theorem 4.3 is established by an exact finite enumeration of all 9841 projective points, with the in-paper verifier in Appendix A implementing the same closure rule and assertions; the rank-distribution check against [3, p. 5] is only an input-transcription sanity check, while the closure statistic is a separate calculation. The proof of Theorem 5.1 then splits into the invertible case, handled by the externally published A-group theorem of Abdollahi, Faghihi, Linton, and O'Brien, and the zero-on-Frattini case, handled internally using P' abelian, the cube relations, and Omega_1(P') = Z(P). None of these inputs is fitted to the target statement, and none is a self-citation by the present authors; the cited structural facts are prior independent results by other authors with stated assumptions that do not include the E-property. The final reliance on the exact execution of the printed verifier is a checkable computational premise, not a circular reduction. No prediction is recovered from fitted values, and no load-bearing self-citation chain appears.
Assumptions & free parameters
assumptions (4)
- domain assumption The group P defined by the presentation in (3) has the structure stated in Proposition 2.1: order 3^84, nilpotency class three, P' = Phi(P), P' abelian, Omega_1(P') = Z(P), and P is an A-group.
- domain assumption The reference verifier in Appendix A correctly implements exact row reduction over F3, the support computation, and the closure process (11).
- standard math Standard facts from finite p-group theory: Burnside's basis theorem, surjective endomorphisms of finite groups are automorphisms, and lower exponent-3 central series subgroups are fully invariant.
- standard math In the class-two quotient P/P_2(P), the cube map descends to an F3-linear map q: V -> Lambda^2 V.
Cite this review
Pith. "Pith review of A Finite E-Group of Nilpotency Class Three." pith.science (2026). https://pith.science/paper/CTT5DG4T
@misc{pith2026260807275,
author = {Pith},
title = {Pith review of: A Finite E-Group of Nilpotency Class Three},
year = {2026},
howpublished = {\url{https://pith.science/paper/CTT5DG4T}},
note = {Machine review of arXiv:2608.07275}
}
abstract
A group is an E-group if every element commutes with each of its endomorphic images. Caranti asked whether a finite E-group can have nilpotency class three. We prove that the $3$-group of order $3^{84}$ introduced by Abdollahi, Faghihi, and Mohammadi Hassanabadi, and later shown by Abdollahi, Faghihi, Linton, and O'Brien to have the corresponding automorphism property, is an E-group. Let $P$ denote this group and put $V=P/\Phi(P)\cong \mathbb{F}_3^9$. The nine power relations of $P$ determine a linear map $q:V\longrightarrow\Lambda^2 V$. We prove that $q$ has no nonzero proper subspace $U$ satisfying $q(U)\subseteq\Lambda^2 U$. Since the image induced by any endomorphism of $P$ on $V$ has precisely this closure property, every endomorphism acts on $V$ either invertibly or trivially. The invertible case is the known A-group case. In the trivial case the image first lies in $\Phi(P)=P'$, and the power relations then force it into $\Omega_1(P')=Z(P)$. Thus every element commutes with every endomorphic image. The tensor rigidity is reduced to an exact finite calculation on the $9841$ points of $\mathrm{PG}(8,3)$.
Reference graph
Works this paper leans on
-
[1]
A. Abdollahi, A. Faghihi, and A. Mohammadi Hassanabadi,Minimal number of generators and minimum order of a non-abelian group whose elements commute with their endomorphic images, Comm. Algebra36 (2008), no. 5, 1976–1987,https://doi.org/10.1080/00927870801941903
-
[2]
A. Abdollahi, A. Faghihi, and A. Mohammadi Hassanabadi,3-generator groups whose elements commute with their endomorphic images are abelian, Comm. Algebra36(2008), no. 10, 3783–3791,https://doi. org/10.1080/00927870802160727
-
[3]
A. Abdollahi, A. Faghihi, S. A. Linton, and E. A. O’Brien,Finite3-groups of class3whose elements commute with their automorphic images, Arch. Math. (Basel)95(2010), 1–7,https://doi.org/10.1007/ s00013-010-0144-y
work page 2010
-
[4]
Caranti,Finite p-groups of exponentp2 in which each element commutes with its endomorphic images, J
A. Caranti,Finite p-groups of exponentp2 in which each element commutes with its endomorphic images, J. Algebra97(1985), no. 1, 1–13,https://doi.org/10.1016/0021-8693(85)90068-7
-
[5]
Caranti,A module-theoretic approach to abelian automorphism groups, Israel J
A. Caranti,A module-theoretic approach to abelian automorphism groups, Israel J. Math.205(2015), no. 1, 235–246,https://doi.org/10.1007/s11856-014-1106-z. 10 XINAN DAI, WENHAO DENG, YINGDONG SHI, TAILIN WU, AND YUCHEN YANG
-
[6]
Caranti,A simple construction for a class ofp-groups with all of their automorphisms central, Rend
A. Caranti,A simple construction for a class ofp-groups with all of their automorphisms central, Rend. Semin. Mat. Univ. Padova135(2016), 251–258,https://doi.org/10.4171/RSMUP/135-14
-
[7]
A module-theoretic approach to abelian automorphism groups
A. Caranti,Erratum to “A module-theoretic approach to abelian automorphism groups”, Israel J. Math. 215(2016), no. 2, 1025–1026,https://doi.org/10.1007/s11856-016-1401-y
-
[8]
Faudree,Groups in which each element commutes with its endomorphic images, Proc
R. Faudree,Groups in which each element commutes with its endomorphic images, Proc. Amer. Math. Soc.27(1971), 236–240,https://doi.org/10.1090/S0002-9939-1971-0269737-3
Show all 12 references
-
[9]
S. P. Glasby, F. A. M. Ribeiro, and C. Schneider,Duality betweenp-groups with three characteristic subgroups and semisimple anti-commutative algebras, Proc. Roy. Soc. Edinburgh Sect. A150(2020), no. 4, 1827–1852,https://doi.org/10.1017/prm.2018.159
2020 doi
-
[10]
E. I. Khukhro and V. D. Mazurov, eds.,Unsolved Problems in Group Theory. The Kourovka Notebook, 21st ed., Novosibirsk, 2026; July 3, 2026 update, Problem 11.46
2026
-
[11]
J. J. Malone,More on groups in which each element commutes with its endomorphic images, Proc. Amer. Math. Soc.65(1977), no. 2, 209–214
1977
-
[12]
Traustason,Symplectic alternating algebras, Internat
G. Traustason,Symplectic alternating algebras, Internat. J. Algebra Comput.18(2008), no. 4, 719–757, https://doi.org/10.1142/S0218196708004585. College of Future Information and Technology, Fudan University, Shanghai, China Current address: Department of Artificial Intelligenc...
2008 doi
Reviewed August 10, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.