Finite E-Group of Nilpotency Class Three Proven to Exist

A new mathematical proof establishes that a finite E-group can have nilpotency class three, resolving a question posed by Caranti. The proof focuses on a specific 3-group of order 3^84, originally introduced by Abdollahi, Faghihi, and Mohammadi Hassanabadi, and later studied by Abdollahi, Faghihi, Linton, and O'Brien for its automorphism properties. The group, denoted P, is shown to be an E-group, meaning every element commutes with each of its endomorphic images. The proof involves analyzing the quotient V = P/Φ(P) ≅ F_3^9, where the nine power relations of P determine a linear map q: V → Λ^2 V. The authors demonstrate that q has no nonzero proper subspace U satisfying q(U) ⊆ Λ^2 U. Since the image induced by any endomorphism of P on V has this closure property, every endomorphism acts on V either invertibly or trivially. The invertible case corresponds to the known A-group case, while in the trivial case, the image lies in Φ(P) = P', and the power relations force it into Ω_1(P'). This result confirms that finite E-groups can indeed have nilpotency class three, answering Caranti's question in the affirmative. The paper is available on arXiv under the identifier 2608.07275.