A Finite E-Group of Nilpotency Class Three

Technical Summary: A Finite E-Group of Nilpotency Class Three

Problem Statement
The paper addresses a specific open question in finite group theory posed by Caranti and recorded as Problem 11.46(a) in the Kourovka Notebook: Does there exist a finite E-group of nilpotency class three?

An E-group is defined as a group $G$ where every element commutes with all of its endomorphic images; formally, $[a, a^\phi] = 1$ for all $a \in G$ and $\phi \in \text{End}(G)$. While the existence of finite E-groups of nilpotency class two is well-established (e.g., by Caranti), the case for class three remained unresolved. The transition from A-groups (where the condition holds only for automorphisms) to E-groups is non-trivial because singular endomorphisms can produce images that are invisible to the automorphism group, requiring specific control over these singular actions.

Methodology and Framework
The authors analyze a specific group $P$ of order $384$ and nilpotency class three, originally introduced by Abdollahi, Faghihi, and Mohammadi Hassanabadi [2] and later studied as an A-group by Abdollahi et al. [3]. The group is a 3-group generated by nine elements $x_1, \dots, x_9$ subject to specific power relations where each cube $x_i^3$ is expressed as a product of four commutators.

The proof strategy relies on reducing the group-theoretic endomorphism problem to a linear algebraic problem over the field $\mathbb{F}_3$:

1. Frattini Quotient: Since $\Phi(P) = P'$, the quotient $V = P/\Phi(P)$ is a 9-dimensional vector space over $\mathbb{F}_3$.
2. Relation Tensor: The nine cube relations of $P$ induce a linear map $q: V \to \Lambda^2 V$.
3. Naturality: For any endomorphism $\phi$ of $P$ inducing a linear map $L$ on $V$, the functoriality of powers and commutators yields the compatibility equation $q \circ L = (\Lambda^2 L) \circ q$.
4. Closure Property: This equation implies that the image of $L$, denoted $U = \text{im } L$, must be q-closed, meaning $q(U) \subseteq \Lambda^2 U$.

The core of the argument is Theorem 1.2 (Relation-tensor rigidity), which asserts that the only subspaces $U \le V$ satisfying $q(U) \subseteq \Lambda^2 U$ are the trivial subspace $\{0\}$ and the entire space $V$.

Key Results and Calculations
The proof of the rigidity theorem involves an exhaustive finite calculation:

• Projective Enumeration: The authors verify the condition for every projective point in $PG(8, 3)$. There are exactly $9841$ such points.
• Closure Algorithm: For each nonzero vector $v$, they construct the least $q$-closed subspace containing $v$ by iteratively adjoining the supports of $q(u)$ for basis elements.
• Verification: Using exact Gaussian elimination over $\mathbb{F}_3$, they demonstrate that for every one of the $9841$ projective points, the resulting closure is the entire space $V$.
• Symmetry Analysis: The authors investigate whether symmetry reduction could simplify this check. They prove (Proposition 4.5) that the linear stabilizer of the tensor $q$ is trivial ($\{I\}$) and its linear similitude group is $\{I, -I\}$. Consequently, the induced action on the projective space is trivial, meaning no non-trivial orbit compression is possible; the exhaustive check of all $9841$ points is necessary.

Proof of the E-Group Property
With the rigidity of $q$ established, the authors analyze the two possible cases for an endomorphism $\phi$ of $P$ based on its induced map $L$ on $V$:

1. Invertible Case ($\text{im } L = V$): By Burnside's Basis Theorem, $\phi$ is surjective and thus an automorphism. Since $P$ is known to be an A-group [3], the condition $[a, a^\phi] = 1$ holds.
2. Trivial Case ($\text{im } L = 0$): Here, $\phi(P) \le \Phi(P) = P'$. Since $P'$ is abelian, applying $\phi$ to the cube relations forces the image of every generator to have order at most 3. Using the identity $\Omega_1(P') = Z(P)$ (from Proposition 2.1), the image is forced into the center $Z(P)$. Thus, $\phi(P) \le Z(P)$, implying $[a, a^\phi] = 1$ trivially.

Significance and Claims
The paper claims to provide a positive answer to Caranti's question, establishing the existence of a finite E-group of nilpotency class three.

• Novelty: The witness group $P$ is not new, but its status as an E-group was previously unknown. The novelty lies in the proof that singular endomorphisms (those inducing the zero map on the Frattini quotient) also satisfy the E-condition.
• Mechanism: The authors highlight that the proof relies on two distinct mechanisms: a linear rigidity (the tensor $q$ admits no proper closed subspaces) and a group-theoretic collapse (the cube relations force the image of a singular endomorphism from the second center $P' = Z_2(P)$ down to the center $Z(P)$).
• Verification: The result is supported by a "finite certificate" (Appendix A) that allows the calculation to be verified independently using only the data presented in the paper, without external files. The calculation is described as exhaustive rather than probabilistic.

The paper situates its work within the broader context of 2-Engel groups and anticommutative algebras, noting that the dual of the relation map defines a simple anticommutative algebra, though the primary contribution remains the specific group-theoretic construction and its verification.