具有高阶仿射映射的有限群
Finite groups with an affine map of large order
摘要：
设$G$是一个群。对于$G$的固定自同构$α$和固定$g\in G$，形式为$x\mapsto x^αg$的函数$G\rightarrow G$称为$G$的仿射映射。本文研究了具有高阶仿射映射的有限群$G$.更确切地说，我们证明了，如果$G$承认一个仿射映射的阶大于$\frac{1}{2}|G|$，那么$G$是可解的导出长度在大多数$3$。我们还表明，更一般地说，对于每个$ρ\in\left(0,1\right]$，如果$G$承认仿射映射的顺序至少$ρ|G|$，那么最大的可解正规子群$G$导出长度最多$4\lfloor\log_2(ρ^{-1})\rfloor+3$。
英文摘要：
Let $G$ be a group. A function $G\rightarrow G$ of the form $x\mapsto x^αg$ for a fixed automorphism $α$ of $G$ and a fixed $g\in G$ is called an affine map of $G$. In this paper, we study finite groups $G$ with an affine map of large order. More precisely, we show that if $G$ admits an affine map of order larger than $\frac{1}{2}|G|$, then $G$ is solvable of derived length at most $3$. We also show that more generally, for each $ρ\in\left(0,1\right]$, if $G$ admits an affine map of order at least $ρ|G|$, then the largest solvable normal subgroup of $G$ has derived length at most $4\lfloor\log_2(ρ^{-1})\rfloor+3$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
