A Finite E-Group of Nilpotency Class Three

2026-08-07

AI Summary

This study resolves an open problem posed by Caranti by constructing and verifying, for the first time, a finite E-group of order \(3^{84}\) and nilpotency class three. The approach involves analyzing a quadratic map \(q: V \to \Lambda^2 V\) on the quotient \(V = P/\Phi(P)\), leveraging tools from exterior algebra, finite field projective geometry (specifically PG(8,3)), and the structure of endomorphisms. By introducing a tensor rigidity condition, the problem is reduced to a finite computation. The results show that every endomorphism of the group acts on \(V\) either invertibly or trivially, and in all cases its image lies within the center \(Z(P)\). Consequently, every element commutes with all of its endomorphic images, thereby satisfying the defining property of an E-group.

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 $V=P/Φ(P)\cong \mathbb{F}_3^9$. The nine power relations of $P$ determine a linear map $q:V\longrightarrowΛ^2 V$. We prove that $q$ has no nonzero proper subspace $U$ satisfying $q(U)\subseteqΛ^2 U$. 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 $Φ(P)=P'$, and the power relations then force it into $Ω_1(P')=Z(P)$. Thus every element commutes with every endomorphic image. The tensor rigidity is reduced to an exact finite calculation on the $9841$ points of $\mathrm{PG}(8,3)$.

Problem

• E-group
• nilpotency class
• finite group
• endomorphism
• power-commuting

Innovation

• E-group
• nilpotency class three
• tensor rigidity
• endomorphic image
• finite p-group