A Finite E-Group of Nilpotency Class Three

A FINITE E-GROUP OF NILPOTENCY CLASS THREE XINAN DAI, WENHAO DENG, YINGDONG SHI, TAILIN WU, AND YUCHEN YANG 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 384 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/Φ(P ) ∼ = F93 . The nine power relations of P determine a linear map

arXiv:2608.07275v1 [math.GR] 7 Aug 2026

q : V −→ Λ2 V. We prove that q has no nonzero proper subspace U satisfying q(U ) ⊆ Λ2 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 Φ(P ) = P ′ , and the power relations then force it into Ω1 (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).

2020 Mathematics Subject Classification. Primary 20D15, 20D45; Secondary 20F45, 17A30. Key words and phrases. E-groups, finite 3-groups, endomorphisms, 2-Engel groups, exterior squares, anticommutative algebras. 1. Introduction For a group G and an endomorphism φ ∈ End(G), write aφ = φ(a). The group G is an E-group if (1)

[a, aφ ] = 1

for every a ∈ G and every φ ∈ End(G). Requiring (1) only for automorphisms gives an A-group. 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. The first nonabelian E-groups were constructed by Faudree [8]; see also Malone’s early analysis of the resulting endomorphism dichotomy [11]. Caranti later gave a systematic classtwo family of finite p-groups of exponent p2 [4], while Abdollahi, Faghihi, and Mohammadi Hassanabadi established sharp restrictions on the number of generators and the order of nonabelian E-groups [1]. Every A-group is 2-Engel and hence nilpotent of class at most three; this elementary but important bound is recalled in [3, p. 1]. Class-two E-groups are therefore well represented in the literature, whereas the existence of a finite E-group of class three remained the question posed by Caranti and recorded as Problem 11.46(a) in the Kourovka Notebook [10, Problem 11.46(a)]. We answer this question positively. Theorem 1.1. There exists a finite E-group of order 384 and nilpotency class three. The witness is not a new group. Abdollahi, Faghihi, and Mohammadi Hassanabadi introduced an explicit nine-generator group P in [2, Remark 2.1], where they recorded its basic structure and left open whether it is an E-group. Abdollahi, Faghihi, Linton, and O’Brien subsequently proved that P is an A-group and determined the central-series data that we shall Xinan Dai is currently a Ph.D. student at Fudan University and a visiting student at the AI for Scientific Simulation and Discovery Lab, Westlake University. Wenhao Deng is a student at the University of Glasgow and is currently an intern at the AI for Scientific Simulation and Discovery Lab, Westlake University. 1

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use below [3, pp. 1–2]. Thus the remaining problem is sharply defined: one must understand the noninvertible endomorphisms of this particular class-three group. The main point of the proof is that the endomorphism problem has a small linear shadow. Since Φ(P ) = P ′ , the Frattini quotient V = P/Φ(P ) is a nine-dimensional vector space over F3 . Passing to the appropriate class-two exponent-3 quotient identifies its commutator layer with Λ2 V . The nine cube relations of P then give a linear map q : V → Λ2 V . If an endomorphism of P induces L ∈ EndF3 (V ), functoriality of powers and commutators yields (2)

q ◦ L = (Λ2 L) ◦ q.

Consequently the subspace U = im L satisfies q(U ) ⊆ Λ2 U . This observation isolates the exact rigidity statement needed in the group argument. Theorem 1.2 (Relation-tensor rigidity). For the relation map q attached to P , the only subspaces U ≤ V such that q(U ) ⊆ Λ2 U are 0 and V . The theorem leaves only two possibilities for an endomorphism of P . If L is onto, Burnside’s basis theorem makes the endomorphism surjective and hence, because P is finite, an automorphism. This is exactly the case settled in [3, Theorem 1.2]. If L = 0, then the image lies in Φ(P ) = P ′ . Here the nilpotency-class-three feature becomes visible: P ′ = Z2 (P ) is strictly larger than Z(P ), so trivial action on the Frattini quotient does not by itself give central image. Applying the endomorphism to all nine cube relations kills their commutator sides, because P ′ is abelian, and forces every generator image to have order at most three. The published identity Ω1 (P ′ ) = Z(P ) then pushes the whole image into the center. This second step is the part that has no analogue to check in the usual special class-two constructions. The proof of Theorem 1.2 is finite but not merely experimental. A bivector ω ∈ Λ2 V has an intrinsic support, the image of the contraction map V ∗ → V . Starting from a nonzero v ∈ V and repeatedly adjoining the supports of q(u) produces the least q-closed subspace containing v. Hence it is enough to test one representative of each point of PG(8, 3), of which there are (39 − 1)/2 = 9841. Exact row reduction over F3 shows that every one of these closures is all of V . As a separate check on the input tensor, the ranks of the associated alternating matrices occur with projective multiplicities 2, 478, 9361 in ranks 4, 6, 8, respectively; doubling these numbers recovers the distribution for the 19682 nonzero vectors recorded in [3, p. 5]. The closure calculation is stronger than that rank distribution and is the new finite certificate used here. The calculation is reproduced inside the paper: Appendix A contains a package-free exact reference verifier whose input is precisely the nine displayed rows of q. Thus the finite step can be checked from the paper alone, without ancillary files. A natural question is whether the 9841 projective points can instead be collapsed to a small number of orbits under symmetries of q. For this particular tensor the answer is no in the natural linear sense. In Section 4.2 we show that its exact linear stabilizer is trivial and that its linear similitude group is {±I}; hence the induced projective action is trivial. The exhaustive projective check is therefore not concealing an unused large symmetry group. There is also a useful algebraic way to read the rigidity. Dualizing q defines an anticommutative multiplication on V ∗ ; under orthogonal complements, q-closed subspaces of V correspond exactly to ideals of this algebra. Theorem 1.2 therefore says that the resulting nine-dimensional anticommutative algebra is simple. This places the calculation near the linear-algebraic methods used by Caranti [5, Section 3] and the group–anticommutative-algebra correspondences studied by Glasby, Ribeiro, and Schneider [9]; symplectic alternating algebras arising from 2-Engel groups provide another neighboring framework [12]. We use these parallels only for interpretation: the group P , its class-three lifting step, and the particular rigidity statement proved here are distinct. Section 2 recalls exactly the published information about P that enters the proof. Section 3 constructs the relation tensor and proves the endomorphism compatibility equation. Section 4

A FINITE E-GROUP OF NILPOTENCY CLASS THREE

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gives the closed-subspace criterion, the finite certificate, and the stabilizer calculation explaining why no nontrivial projective orbit reduction is available. Section 5 proves Theorem 1.1; Section 6 records the dual-algebra interpretation and the precise scope of the argument. Appendix A contains the complete reference verifier. 2. The group and its first exponent-3 quotient We use the commutator convention [x, y] = x−1 y −1 xy. Let P be the largest 2-Engel group of exponent 27 generated by x1 , . . . , x9 , subject to x31 = [x2 , x3 ][x4 , x5 ][x6 , x7 ][x8 , x9 ], x32 = [x1 , x3 ][x4 , x6 ][x5 , x8 ][x7 , x9 ], x33 = [x1 , x2 ][x4 , x7 ][x5 , x9 ][x6 , x8 ], x34 = [x1 , x5 ][x2 , x6 ][x3 , x9 ][x7 , x8 ], x35 = [x1 , x4 ][x2 , x8 ][x3 , x7 ][x6 , x9 ],

(3)

x36 = [x1 , x7 ][x2 , x9 ][x3 , x5 ][x4 , x8 ], x37 = [x1 , x8 ][x4 , x9 ][x3 , x6 ][x2 , x5 ], x38 = [x1 , x9 ][x3 , x4 ][x2 , x7 ][x5 , x6 ], x39 = [x1 , x6 ][x3 , x8 ][x2 , x4 ][x5 , x7 ]. Each unordered pair {i, j} with 1 ≤ i < j ≤ 9 occurs exactly once on the right-hand sides. The presentation and the following structural data come from [2, Remark 2.1] and the consistent presentation in [3, p. 2]. Proposition 2.1 (Published structure). The group P has order 384 and nilpotency class three. With Pi (P ) denoting the lower exponent-3 central series,

(4)

exp(P/P ′ ) = 3, P ′ = Z2 (P ) ∼ = C 36 × C 3 , 9

3

|P/P2 (P )| = 345 , ∼ C 39 . Ω1 (P ′ ) = γ3 (P ) = Z(P ) = 3

Moreover P is an A-group. Since P is a finite 3-group, Φ(P ) = P 3 P ′ . The equality exp(P/P ′ ) = 3 therefore gives (5)

Φ(P ) = P ′ .

In particular, V := P/Φ(P ) ∼ = F93 , with basis ei = xi Φ(P ). The direct-product description in (4) also shows that P ′ is abelian. Put P = P/P2 (P ). Because P2 (P ) ≤ Φ(P ), P still needs nine generators, so |Φ(P )| = 345−9 = 336 . The group P has class at most two, its derived subgroup has exponent three, and its ′ abelianization has exponent three. Hence Φ(P ) = P . The 36 commutators [xi , xj ]P2 (P ) with ′ i < j generate P and have order at most three; the order calculation above therefore shows that they form an F3 -basis. We obtain a canonical identification, relative to the chosen basis of V , (6)

[xi , xj ]P2 (P ) ←→ ei ∧ ej ,

1 ≤ i < j ≤ 9.

The class-two computation of Abdollahi–Faghihi–Linton–O’Brien also gives |Z(P )| = 336 [3, ′ Lemma 3.1 and p. 5]. Since P = Φ(P ) already has this order and is central, we shall use (7)

′

Z(P ) = Φ(P ) = P .

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XINAN DAI, WENHAO DENG, YINGDONG SHI, TAILIN WU, AND YUCHEN YANG

3. The relation tensor In a class-two group whose commutator subgroup has exponent three, (xy)3 = x3 y 3 . Thus cubing in P descends to an F3 -linear map q : V −→ Λ2 V.

(8) Using (6), the relations (3) give

q(e1 ) = e2 ∧ e3 + e4 ∧ e5 + e6 ∧ e7 + e8 ∧ e9 , q(e2 ) = e1 ∧ e3 + e4 ∧ e6 + e5 ∧ e8 + e7 ∧ e9 , q(e3 ) = e1 ∧ e2 + e4 ∧ e7 + e5 ∧ e9 + e6 ∧ e8 , q(e4 ) = e1 ∧ e5 + e2 ∧ e6 + e3 ∧ e9 + e7 ∧ e8 , q(e5 ) = e1 ∧ e4 + e2 ∧ e8 + e3 ∧ e7 + e6 ∧ e9 , q(e6 ) = e1 ∧ e7 + e2 ∧ e9 + e3 ∧ e5 + e4 ∧ e8 , q(e7 ) = e1 ∧ e8 + e4 ∧ e9 + e3 ∧ e6 + e2 ∧ e5 , q(e8 ) = e1 ∧ e9 + e3 ∧ e4 + e2 ∧ e7 + e5 ∧ e6 , q(e9 ) = e1 ∧ e6 + e3 ∧ e8 + e2 ∧ e4 + e5 ∧ e7 . This is the sole tensor used below. Lemma 3.1 (Naturality). Let φ ∈ End(P ) and let L ∈ EndF3 (V ) be the induced linear map. Then (9)

q ◦ L = (Λ2 L) ◦ q.

Consequently (10)

q(im L) ⊆ Λ2 (im L).

Proof. The subgroup P2 (P ) is fully invariant, so φ induces an endomorphism of P . Let v ∈ V and choose a lift y ∈ P . Modulo P2 (P ) the cube of y represents q(v). Applying φ and using [a, b]φ = [aφ , bφ ] gives two descriptions of the same cube: q(Lv) and (Λ2 L)q(v). This proves (9). Its right-hand side belongs to Λ2 (im L), which gives (10). □ Definition 3.2. A subspace U ≤ V is q-closed if q(U ) ⊆ Λ2 U . The point of Definition 3.2 is not to classify all endomorphisms: Lemma 3.1 only says that every induced endomorphism image is q-closed. This one-way implication is exactly what is needed to rule out singular actions on the Frattini quotient. 4. Closed subspaces and rigidity For ω ∈ Λ2 V , contraction identifies ω with an alternating map V ∗ → V . Define ω

supp(ω) = im(V ∗ −−→ V ). Equivalently, after choosing the basis e1 , . . . , e9 , supp(ω) is the column space of the skew matrix representing ω. Lemma 4.1 (Support criterion). For U ≤ V and ω ∈ Λ2 V , ω ∈ Λ2 U

⇐⇒

supp(ω) ≤ U.

Proof. Choose a basis of U and extend it to one of V . If ω ∈ Λ2 U , its skew matrix has nonzero entries only in the U × U block, so its image lies in U . Conversely, if its image lies in U , all rows outside the U -block vanish; skew-symmetry forces the corresponding columns to vanish as well. Thus no exterior coordinate of ω involves a basis vector outside U , which is precisely ω ∈ Λ2 U . □

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Starting from 0 ̸= v ∈ V , define U0 (v) = ⟨v⟩. Given Ur (v), choose a basis Br and set (11)

X

Ur+1 (v) = Ur (v) +

supp(q(u)).

u∈Br

The process stabilizes because dim V = 9. Lemma 4.2 (Least closed subspace). The stable value clq (v) of (11) is independent of the chosen bases and is the least q-closed subspace containing v. Proof. If u is a linear combination of the vectors in Br , then q(u) is the same linear combination of their images, and the support of a sum of bivectors is contained in the sum of the individual supports. Hence adjoining the supports for a basis is equivalent to adjoining them for all vectors in Ur (v); this proves basis independence. At a fixed point, Lemma 4.1 gives q(Ur (v)) ⊆ Λ2 Ur (v), so the stable subspace is q-closed. If W is any q-closed subspace containing v, then Lemma 4.1 shows inductively that Ur (v) ≤ W for every r. Therefore the stable value is the least such subspace. □ 4.1. The exhaustive finite certificate. Scalar multiples have the same closure, so only projective directions must be checked. Normalize each nonzero vector by making its first nonzero coordinate equal to 1. The number of normalized vectors is 39 − 1 = 9841. 3−1 For each normalized v, form the alternating matrix Q(v) representing q(v) and iterate (11), using exact Gaussian elimination over F3 for all spans and supports. The exhaustive calculation gives:

(12)

| PG(8, 3)| =

Table 1. Exact projective verification for the relation tensor. rank 4 rank 6 rank 8 Projective points [v] Nonzero vectors v

2 4

478 956

9361 18722

Closure statistic count dim clq (v) = 9 dim clq (v) < 9

9841 0

A finer audit trail records the complete dimension growth of the closure. Writing, for example, (1, 8, 9) when the successive dimensions are 1, 8, 9, the exact profile is (13)

profile (1, 9) (1, 8, 9) (1, 7, 9) (1, 6, 9) (1, 5, 9) (1, 4, 9) count 6138 3223 426 52 1 1.

In particular every projective point reaches V after at most two support enlargements. The two rank-four directions are represented by (14)

(0, 1, 1, 1, 1, 1, 1, 1, 1),

(1, 1, 1, 1, 1, 1, 1, 0, 1),

and have growth profiles (1, 4, 9) and (1, 5, 9), respectively. 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. The closure data are a different computation: they record the iterative support condition of Lemma 4.2, not merely the rank of q(v). Theorem 4.3 (Tensor rigidity). The only q-closed subspaces of V are 0 and V . Proof. Let U be nonzero and q-closed. Choose 0 ̸= v ∈ U . Lemma 4.2 gives clq (v) ≤ U . By (12) and the exhaustive calculation in Table 1, clq (v) = V for every nonzero projective direction. Hence U = V . The zero subspace is trivially q-closed. □

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XINAN DAI, WENHAO DENG, YINGDONG SHI, TAILIN WU, AND YUCHEN YANG

Remark 4.4 (What the computation certifies). The finite step is exhaustive, not probabilistic: Lemma 4.2 reduces every nonzero q-closed subspace to the closure of one projective point, and (12) lists all such points. A direct exact check of the 36 commutator pairs gives 36/36 distinct pairs. The reference verifier in Appendix A reproduces Table 1 and (13) using only integer arithmetic modulo three. As a negative control, deleting the first row q(e1 ) leaves ⟨e1 ⟩ closed, so the same procedure correctly detects a deliberately introduced proper closed subspace. 4.2. The stabilizer and the orbit question. There is an obvious possible shortcut to an exhaustive projective verification: compute a symmetry group of q and check one point in each orbit. In the present example the natural symmetry group is too small for this strategy. Proposition 4.5 (Trivial projective tensor symmetry). Set Γ(q) = {g ∈ GL(V ) : qg = (Λ2 g)q} and let Γsim (q) = {g ∈ GL(V ) : qg = λ(Λ2 g)q for some λ ∈ F× 3 }. Then Γ(q) = {I},

Γsim (q) = {I, −I}.

Consequently the induced action of the natural linear similitude group on PG(V ) = PG(8, 3) is trivial. Proof. Let g ∈ Γ(q). The class-two group P has Frattini quotient V , commutator layer Λ2 V via (6), and power map q. These data give a full class-two presentation of P . Indeed, introduce central symbols cij of order three for 1 ≤ i < j ≤ 9, impose [xi , xj ] = cij and the nine cube relations encoded by q, and kill all triple commutators. Every element of the resulting group has a normal form Y bij xa11 · · · xa99 cij , 0 ≤ ai , bij < 3, i<j

so its order is at most 39+36 .

It maps onto P , which has exactly that order; hence the presentation is exact. The assignments on V and Λ2 V given by g and Λ2 g therefore preserve the defining relations exactly when qg = (Λ2 g)q. Thus g lifts to an endomorphism of P . Since g is invertible on the Frattini quotient, Burnside’s basis theorem makes this lift an automorphism. Abdollahi–Faghihi–Linton–O’Brien proved Aut(P ) = Autc (P ) [3, Lemma 3.1, p. 5]. By (7), a central automorphism of P acts trivially on P /Φ(P ) = V . Therefore g = I, proving Γ(q) = {I}. Now suppose g ∈ Γsim (q), so qg = λ(Λ2 g)q with λ ∈ F× 3 = {1, −1}. Put h = λg. Since 2 λ = 1 and q is linear, qh = λqg = λ2 (Λ2 g)q = (Λ2 h)q. Thus h ∈ Γ(q), so h = I and g = λI. Conversely I and −I plainly satisfy the corresponding similitude identities. Both scalars induce the identity on projective space. □ Remark 4.6. Proposition 4.5 is not needed for the E-group theorem. Its purpose is to explain the finite proof. It rules out the most natural orbit compression: the tensor itself has no nontrivial projective linear symmetry with which to identify different directions. It does not rule out a completely different, non-symmetry-based argument for simplicity of the dual algebra. For the present certificate, however, the projective enumeration is already orbit-minimal for the natural tensor similitude group. 5. Proof of the E-group theorem 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.

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Proof. Fix φ ∈ End(P ) and let L be its induced map on V = P/Φ(P ). By Lemma 3.1, im L is q-closed. Theorem 4.3 gives (15)

im L = V

or

im L = 0.

Suppose first that im L = V . Then φ(P )Φ(P ) = P . Burnside’s basis theorem implies φ(P ) = P , and a surjective endomorphism of a finite group is an automorphism. Proposition 2.1 therefore gives [a, aφ ] = 1 for every a ∈ P . Suppose now that im L = 0. By (5), φ(P ) ≤ Φ(P ) = P ′ . The group P ′ is abelian. Apply φ to each of the nine relations in (3). Every commutator on a right-hand side becomes a commutator of two elements of P ′ and hence is trivial. Therefore φ(xi )3 = 1

(1 ≤ i ≤ 9).

Since each φ(xi ) also lies in P ′ , the identity Ω1 (P ′ ) = Z(P ) from Proposition 2.1 yields φ(xi ) ∈ Z(P ) for every i. The xi generate P , so φ(P ) ≤ Z(P ). Again [a, aφ ] = 1 for every a ∈ P. The two cases in (15) cover all endomorphisms. The order and exact nilpotency class are part of Proposition 2.1. □ Remark 5.2. The second case does not say that an endomorphism inducing zero on P/Φ(P ) is the zero endomorphism. It first allows an arbitrary image inside P ′ = Z2 (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. 6. Algebraic meaning and scope The condition in Theorem 4.3 has a useful dual form. Let A = V ∗ and define a bilinear anticommutative product by (16)

⟨α · β, v⟩ = ⟨α ∧ β, q(v)⟩,

α, β ∈ V ∗ , v ∈ V.

For U ≤ V , write U ⊥ = {α ∈ V ∗ : α(U ) = 0}. 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. Proof. The subspace U ⊥ is an ideal precisely when, for every α ∈ U ⊥ , β ∈ V ∗ , and u ∈ U , 0 = ⟨α · β, u⟩ = ⟨α ∧ β, q(u)⟩. For a fixed u, these equalities for all α ∈ U ⊥ and β ∈ V ∗ are equivalent to q(u) ∈ Λ2 U . Hence the ideal condition is equivalent to q(U ) ⊆ Λ2 U . Orthogonal complementation reverses 0 and V , so the absence of nonzero proper q-closed subspaces is exactly the absence of nonzero proper ideals in A. □ This reformulation helps locate the rigidity in a broader pattern. In Caranti’s moduletheoretic treatment of class-two groups, power relations are likewise encoded by a linear map involving Λ2 V [5, Section 3]; the published erratum should be read together with that paper for its later modified constructions [7]. Caranti’s subsequent construction gives examples where the compatible endomorphisms are reduced to the trivial map and the identity [6, Section 4]. The present argument has a similar linear silhouette but a different endpoint: the group is already fixed, the tensor is the one forced by (3), and a zero action on the Frattini quotient still has to be lifted through the noncentral group P ′ = Z2 (P ). 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/Φ(P ). The second is genuinely group-theoretic: once the induced map vanishes, the nine cube relations collapse the image from P ′ to Ω1 (P ′ ) = Z(P ). Neither statement alone proves the E-property. Their combination is what makes the classthree candidate work.

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Appendix A. A self-contained exact reference verifier This appendix makes the finite certificate in Section 4.1 reproducible from the paper alone. The program below uses no external package. Its list R is the zero-based encoding of the four exterior pairs in each of the nine rows of q displayed in Section 3. The routine support computes the column space of the corresponding alternating matrix, closure implements (11), and points enumerates exactly the first-nonzero-coordinate-one representatives of PG(8, 3). All row reduction is performed over F3 by integer arithmetic modulo three. Listing 1. Exact reference verifier for the relation tensor. from collections import Counter from itertools import product p, n = 3, 9 R = ( ((1,2),(3,4),(5,6),(7,8)), ((0,2),(3,5),(4,7),(6,8)), ((0,1),(3,6),(4,8),(5,7)), ((0,4),(1,5),(2,8),(6,7)), ((0,3),(1,7),(2,6),(5,8)), ((0,6),(1,8),(2,4),(3,7)), ((0,7),(3,8),(2,5),(1,4)), ((0,8),(2,3),(1,6),(4,5)), ((0,5),(2,7),(1,3),(4,6)), ) def rref(rows): A = [[x % p for x in row] for row in rows if any(x % p for x in row)] r = 0 for c in range(n): s = next((i for i in range(r, len(A)) if A[i][c]), None) if s is None: continue A[r], A[s] = A[s], A[r] if A[r][c] == 2: A[r] = [(2*x) % p for x in A[r]] for i in range(len(A)): if i != r and A[i][c]: a = A[i][c] A[i] = [(x-a*y) % p for x,y in zip(A[i],A[r])] r += 1 return A[:r] def qmat(v, rel=R): Q = [[0]*n for _ in range(n)] for a, pairs in enumerate(rel): for i,j in pairs: Q[i][j] = (Q[i][j] + v[a]) % p Q[j][i] = (Q[j][i] - v[a]) % p return Q def support(v, rel=R): Q = qmat(v, rel) return rref([[Q[i][j] for i in range(n)] for j in range(n)]) def closure(v, rel=R): B = rref([v]) dims = [len(B)] while True: C = rref(B + [w for u in B for w in support(u, rel)]) if len(C) == len(B): return B, tuple(dims) B = C dims.append(len(B)) def points():

A FINITE E-GROUP OF NILPOTENCY CLASS THREE

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for v in product(range(p), repeat=n): if v == (0,)*n: continue if next(x for x in v if x) == 1: yield v pts = list(points()) assert len(pts) == (p**n-1)//(p-1) == 9841 assert {e for row in R for e in row} == { (i,j) for i in range(n) for j in range(i+1,n)} ranks, closes, profiles = Counter(), Counter(), Counter() for v in pts: ranks[len(support(v))] += 1 B, profile = closure(v) closes[len(B)] += 1 profiles[profile] += 1 assert ranks == Counter({4:2, 6:478, 8:9361}) assert closes == Counter({9:9841}) assert profiles == Counter({ (1,9):6138, (1,8,9):3223, (1,7,9):426, (1,6,9):52, (1,5,9):1, (1,4,9):1}) bad = list(R) bad[0] = () assert len(closure((1,0,0,0,0,0,0,0,0), bad)[0]) == 1 print("9841 projective points; rank profile 4^2, 6^478, 8^9361") print("all closures have dimension 9; negative control detected")

The output is 9841 projective points; rank profile 4^2, 6^478, 8^9361 all closures have dimension 9; negative control detected 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. Statement on computational and writing assistance The TARS agent system assisted with exploratory derivations. The exact finite verification in Section 4.1 was checked with independent Python and C++ implementations over F3 ; the compact Python verifier used for the final certificate is reproduced in Appendix A. The authors are responsible for the mathematical statements, literature attributions, and final text. References [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. Algebra 36 (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. Algebra 36 (2008), no. 10, 3783–3791, https://doi. org/10.1080/00927870802160727. [3] A. Abdollahi, A. Faghihi, S. A. Linton, and E. A. O’Brien, Finite 3-groups of class 3 whose elements commute with their automorphic images, Arch. Math. (Basel) 95 (2010), 1–7, https://doi.org/10.1007/ s00013-010-0144-y. [4] A. Caranti, Finite p-groups of exponent p2 in which each element commutes with its endomorphic images, J. Algebra 97 (1985), no. 1, 1–13, https://doi.org/10.1016/0021-8693(85)90068-7. [5] 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.

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XINAN DAI, WENHAO DENG, YINGDONG SHI, TAILIN WU, AND YUCHEN YANG

[6] A. Caranti, A simple construction for a class of p-groups with all of their automorphisms central, Rend. Semin. Mat. Univ. Padova 135 (2016), 251–258, https://doi.org/10.4171/RSMUP/135-14. [7] 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] 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. [9] S. P. Glasby, F. A. M. Ribeiro, and C. Schneider, Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 4, 1827–1852, https://doi.org/10.1017/prm.2018.159. [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. [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. [12] 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 Intelligence, School of Engineering, Westlake University, Hangzhou, China Email address: xndai23@m.fudan.edu.cn University of Glasgow, Glasgow, United Kingdom Current address: Department of Artificial Intelligence, School of Engineering, Westlake University, Hangzhou, China Email address: dengwenhao@westlake.edu.cn School of Information Science and Technology, ShanghaiTech University, Shanghai, China Email address: shiyd2023@shanghaitech.edu.cn Department of Artificial Intelligence, School of Engineering, Westlake University, Hangzhou, China Email address: wutailin@westlake.edu.cn Department of Artificial Intelligence, School of Engineering, Westlake University, Hangzhou, China Email address: yangyuchen@westlake.edu.cn

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