A Finite E-Group of Nilpotency Class Three

Problem addressed. The paper answers a question posed by Caranti, recorded as Problem 11.46(a) in the Kourovka Notebook: whether a finite E-group can have nilpotency class three. A group is an E-group if every element commutes with each of its endomorphic images, i.e., [a, a phi] = 1 for every a in G and every phi in End(G). Requiring this only for automorphisms gives an A-group. The authors note that The passage from A-groups to E-groups is not formal: singular endomorphisms can have images that are invisible to the automorphism group, and controlling those images is the central issue here.

Main theorem and witness group. The main result is:

Theorem 1.1. There exists a finite E-group of order 384 and nilpotency class three.

The witness is not a new group. It is the 3-group P of order 384 introduced by Abdollahi, Faghihi, and Mohammadi Hassanabadi in [2, Remark 2.1], which was later shown by Abdollahi, Faghihi, Linton, and O'Brien to be an A-group with certain central-series data. The remaining problem was sharply defined: one must understand the noninvertible endomorphisms of this particular class-three group.

Group structure. The group P is the largest 2-Engel group of exponent 27 generated by x 1,, x 9, subject to nine cube relations (displayed in Section 2), where "each unordered pair i,j with 1 i < j 9 occurs exactly once on the right-hand sides." The published structure is recorded as:

Proposition 2.1. The group P has order 3 8 and nilpotency class three. With P i(P) denoting the lower exponent-3 central series,

(P/P') = 3, P' = Z 2(P) C 3 6 times C 3 9, P/P 2(P) = 3 45, 1(P') = gamma 3(P) = Z(P) C 3 9.

Moreover P is an A-group.

Since P is a finite 3-group, (P) = P cubed P'. The equality (P/P') = 3 gives (P) = P', so the Frattini quotient is V = P/ (P) F 3 9, with basis e i = x i (P). The paper also establishes that for the class-two quotient = P/P 2(P), there is a canonical identification [x i, x j]P 2(P) e i e j for 1 i < j 9, identifying the commutator layer with squared V.

Relation tensor. Cubing in descends to an F 3-linear map q: V to squared V. Using the nine relations (3), the tensor q is displayed explicitly as nine rows; for example, q(e 1) = e 2 e 3 + e 4 e 5 + e 6 e 7 + e 8 e 9, and similarly for the other eight basis vectors. This is the sole tensor used below.

The key naturality observation is:

Lemma 3.1 (Naturality). Let phi in End(P) and let L in End F 3(V) be the induced linear map. Then q L = (squared L) q. Consequently q(im, L) 2(im, L).

This motivates the definition: a subspace U V is q-closed if q(U) squared U. The implication is one-way — every induced endomorphism image is q-closed — but this one-way implication is exactly what is needed to rule out singular actions on the Frattini quotient.

Closed-subspace criterion. For omega in squared V, contraction identifies omega with an alternating map V* to V, and the support of omega is its image. The paper proves:

Lemma 4.1 (Support criterion). For U V and omega in squared V, omega in squared U supp(omega) U.

Starting from a nonzero v in V, the paper defines an iterative closure process U r+1(v) = U r(v) + sum u in B r supp(q(u)), where B r is a basis of U r(v). The stable value cl q(v) is:

Lemma 4.2 (Least closed subspace). The stable value cl q(v) of (11) is independent of the chosen bases and is the least q-closed subspace containing v.

Finite certificate. Since scalar multiples have the same closure, only projective directions need checking. The number of normalized vectors (first nonzero coordinate equal to 1) is (3 9 - 1)/(3-1) = 9841 = PG(8,3). The exhaustive calculation gives:

Table 1. Exact projective verification for the relation tensor.

• rank 4: 2 projective points, 4 nonzero vectors

• rank 6: 478 projective points, 956 nonzero vectors

• rank 8: 9361 projective points, 18722 nonzero vectors

• Closure statistic: dim cl q(v) = 9 count 9841; dim cl q(v) < 9 count 0.

The dimension growth profile shows that every projective point reaches V after at most two support enlargements: the profiles are (1,9) with count 6138, (1,8,9) with 3223, (1,7,9) with 426, (1,6,9) with 52, (1,5,9) with 1, and (1,4,9) with 1. The paper notes that "The first line of Table 1 doubles to the published rank distribution for all nonzero vectors in the corresponding nine-dimensional relation space [3, p. 5]. This is an independent check that the tensor has been transcribed correctly."

This yields the central rigidity theorem:

Theorem 4.3 (Tensor rigidity). The only q-closed subspaces of V are 0 and V.

A negative control is included: deleting the first row q(e 1) leaves e 1 closed, so the same procedure correctly detects a deliberately introduced proper closed subspace.

Stabilizer computation. The paper addresses the natural question of whether the 9841 projective points can be collapsed to a small number of orbits under symmetries of q:

Proposition 4.5 (Trivial projective tensor symmetry). Set (q) = g in GL(V): qg = (squared g)q and sim(q) = g in GL(V): qg = lambda(squared g)q for some lambda in F 3 times. Then (q) = I, sim(q) = I, -I. Consequently the induced action of the natural linear similitude group on PG(V) = PG(8,3) is trivial.

The proof uses the fact that the data (V, squared V, q) give a full class-two presentation of, so elements of (q) lift to automorphisms of P; since Aut(P) = Aut c(P) and a central automorphism acts trivially on V, one gets g = I. The authors stress: The exhaustive projective check is therefore not concealing an unused large symmetry group.

Proof of the E-group theorem. The main proof splits into two cases according to (15):

Theorem 5.1. The group P is a finite E-group of nilpotency class three. In particular, Problem 11.46(a) of the Kourovka Notebook has an affirmative answer.

Case 1: im, L = V. Then phi(P) (P) = P, so by Burnside's basis theorem phi(P) = P, and a surjective endomorphism of a finite group is an automorphism. The known A-group property of P gives [a, a phi] = 1.

Case 2: im, L = 0. Then phi(P) (P) = P'. Applying phi to each of the nine relations kills every commutator side because P' is abelian, forcing phi(x i) cubed = 1. Since each phi(x i) in P', the identity 1(P') = Z(P) pushes the entire image into the center, giving [a, a phi] = 1.

The authors emphasize the two-step structure: "The second case does not say that an endomorphism inducing zero on P/ (P) is the zero endomorphism. It first allows an arbitrary image inside P' = Z 2(P) and then uses the cube relations to force that image into the proper subgroup Z(P). Keeping these two steps separate is essential in class three."

Dual algebra interpretation. Dualizing q defines an anticommutative multiplication on A = V* by alpha times beta, v = alpha beta, q(v). Then:

Proposition 6.1. A subspace U V is q-closed if and only if U is an ideal of the anticommutative algebra A. Consequently Theorem 4.3 is equivalent to simplicity of A.

This places the result near the linear-algebraic methods of Caranti and the group–anticommutative-algebra correspondences of Glasby, Ribeiro, and Schneider, though the group P, its class-three lifting step, and the particular rigidity statement proved here are distinct.

Conclusion on the two rigidity mechanisms. The paper concludes: "The conclusion therefore rests on two separate rigidity mechanisms. The first is linear: the relation tensor admits no nonzero proper closed subspace, so a singular endomorphism cannot retain any nonzero direction in P/ (P). The second is genuinely group-theoretic: once the induced map vanishes, the nine cube relations collapse the image from P' to 1(P') = Z(P). Neither statement alone proves the E-property. Their combination is what makes the class-three candidate work."

Verification appendix. Appendix A contains a complete, package-free exact reference verifier in Python whose input is precisely the nine displayed rows of q. The paper states: "This appendix makes the finite certificate in Section 4.1 reproducible from the paper alone... The assertions implement the certificate: a failed pair partition, projective count, rank distribution, closure count, growth profile, or negative control stops execution. In particular, the mathematical input, the exhaustion rule, and an executable implementation all appear in the PDF. It also notes that The exact finite verification in Section 4.1 was checked with independent Python and C++ implementations over F 3."

Based on this paper, here are specific improvements to AI systems for mathematics and automated theorem proving:

• Generate finite certificates for infinite-sounding algebraic claims.

The improved AI system can take a statement like “there exists a finite E-group of class three” and automatically reduce it to a finite linear-algebraic certificate — here a 9-dimensional tensor q: V to squared V over F 3 — together with an exhaustive enumeration plan (9841 projective points). It can then output a self-contained exact verifier script that checks every assertion, including the projective count, rank distribution, closure count, growth profile, and negative control.

• Automatically derive closure criteria and rigid subspace proofs.

Given a tensor q, the AI system can formulate and prove lemmas such as the support criterion omega in squared U supp(omega) U, implement the iterative closure algorithm, and compute the least q-closed subspace containing each vector. It can then report the dimension-growth profiles — e.g. (1,9), (1,8,9), (1,7,9), (1,4,9) — as a fingerprint of the tensor.

• Use rigidity to reduce endomorphism analysis to two boundary cases.

The AI system can reason that, because the only q-closed subspaces are 0 and V, any induced endomorphism L: V to V satisfying q L = squared L q must have image either V or 0. It can then apply Burnside’s basis theorem in the surjective case and, in the zero case, use the nine cube relations plus 1(P') = Z(P) to force the image into the center. This two-step pattern becomes a reusable tactic for proving E-properties in finite p-groups.

• Compute and exploit stabilizer groups to validate exhaustive searches.

The improved AI system can compute the full tensor stabilizer (q) = g: qg = squared g, q and its similitude version, and determine whether the induced action on projective space is trivial. This confirms that the exhaustive check over 9841 points is not hiding a large symmetry orbit collapse, making the brute-force verification logically robust.

• Translate rigidity into dual algebraic simplicity.

The AI system can dualize the tensor q into an anticommutative algebra A = V* and automatically prove that q-closed subspaces correspond exactly to ideals of A. It can then cross-check the rigidity theorem by independently verifying that A is simple, using algorithms for ideals in finite-dimensional algebras.

• Include negative controls to validate the verification pipeline.

The AI system can generate deliberately mutated variants — such as deleting one row q(e 1) — and run the same closure procedure to confirm that a proper closed subspace appears. This mutation-testing step ensures the exhaustive verifier is sensitive and that the original positive result is not vacuous.

• Cross-validate with multiple independent implementations.

The AI system can implement the finite-field verification in both Python and C++ using separate algorithms, then require exact agreement on all counts: 2/478/9361 rank classes, 9841 closure points, and all growth-profile counts. This protects against implementation-specific bugs and transcription errors.

• Automate transcription sanity checks against published data.

When reading a tensor from a paper, the AI system can compute the rank distribution for all nonzero vectors and compare it to previously published distributions (e.g., doubling the projective rank counts). This provides an early automated check that the tensor was transcribed correctly from the original group presentation.

• Produce structured, human-readable proof explanations.

The improved AI system can write a proof that separates the two rigidity mechanisms — the linear rigidity of the tensor and the group-theoretic collapse from P' to Z(P) — making clear why neither alone suffices and how their combination proves the theorem. This supports interpretable AI-generated mathematics.

• Reuse the method as a general tool for finite p-group endomorphism problems.

The AI system can generalize the pipeline: given a finite p-group presentation, construct the cubing tensor q on the Frattini quotient, compute closed subspaces, check rigidity, and classify singular endomorphisms. This makes the technique applicable to other questions in the Kourovka Notebook and beyond.

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/ (P) F 3 9. The nine power relations of P determine a linear map q:V squared V. We prove that q has no nonzero proper subspace U satisfying q(U) squared 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 (P)=P', and the power relations then force it into 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 PG(8,3).