A Finite E-Group of Nilpotency Class Three

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/Phi(P)cong mathbbF39. The nine power relations of P determine a linear map q:VlongrightarrowLambda2 V. We prove that q has no nonzero proper subspace U satisfying q(U)subseteqLambda2 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', and the power relations then force it into Omega1(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).