A Finite E-Group of Nilpotency Class Three

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 $q:V⟶{\Lambda }^{2}V$. 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 $\Phi (P)=P^{\prime }$, and the power relations then force it into ${\Omega }_1(P^{\prime })=Z(P)$. Thus every element commutes with every endomorphic image. The tensor rigidity is reduced to an exact finite calculation on the $9841$ points of $\mathrm{PG}(8,3)$.

9 pages