构造出一个幂零类为三的有限E-群，解决了一个长期未解问题。
A Finite E-Group of Nilpotency Class Three
- 通过分析9维向量空间上的线性映射，证明该群满足端同态像与元素交换的性质。
- 验证了该3-群在模其弗拉贝尼乌斯子群后，所有端同态作用要么可逆要么平凡。
- 适用于代数结构、群论和有限群计算方向的研究者。
若群中每个元素与其自身所有端同态像均交换，则称其为E-群。Caranti提出：有限E-群能否具有幂零类三？本文证明：由Abdollahi、Faghihi和Mohammadi Hassanabadi引入的3-群（阶为3^84），经Abdollahi、Faghihi、Linton和O'Brien证实具有自同构性质，确为一E-群。设P为此群，令V = P/Φ(P) ≅ F₃⁹。P的九个幂关系定义了从V到Λ²V的线性映射q。我们证明：q不存在非零真子空间U满足q(U) ⊆ Λ²U。由于任意自同态在V上的像均具备此闭包性质，故每个自同态在V上或可逆或平凡。可逆情形即已知的A-群情况；平凡情形下，像首入Φ(P) = P′，再由幂关系迫使进入Ω₁(P′) = Z(P)。因此每个元素都与每个端同态像交换。张量刚性化归为对PG(8,3)上9841个点的精确有限计算。
原文摘要 · Abstract (English)
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 $\mathrm{PG}(8,3)$.
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