一类幂零类为3的有限E-群
A Finite E-Group of Nilpotency Class Three
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中文总结 AI 辅助
本文证明阶为3^84的3-群是E-群，解答了Caranti关于有限E-群能否为幂零类3的问题，其自同态作用在商空间上仅可逆或平凡，张量刚性可简化为PG(8,3)点的有限计算。
中文摘要 AI 辅助
群被称为E-群，当且仅当每个元素都与其所有自同态像可交换。Caranti提出问题：有限E-群能否具有幂零类3？我们证明，由Abdollahi、Faghihi和Mohammadi Hassanabadi引入的阶为3^84的3-群（后经Abdollahi、Faghihi、Linton和O'Brien证明具有相应自同构性质）是一个E-群。记该群为P，令V=P/Φ(P)≅𝔽₃⁹，P的9个幂关系确定了线性映射q:V→Λ²V。我们证明q不存在非零真子空间U满足q(U)⊆Λ²U。由于P的任意自同态在V上诱导的像恰好具有该闭包性质，故每个自同态在V上的作用要么可逆，要么平凡：可逆情形是已知的A-群情形；平凡情形下像先包含于Φ(P)=P'，再由幂关系强制进入Ω₁(P')=Z(P)，因此每个元素都与所有自同态像可交换。张量刚性可简化为对PG(8,3)的9841个点的精确有限计算。
英文摘要
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 $\mathrm{PG}(8,3)$.
发表机构
- Westlake University（西湖大学）
- University of Glasgow（格拉斯哥大学）
- School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）
机构由 AI 辅助整理，请以论文原文为准。
