论文标题
有限型组有大量的仿射图
Finite groups with an affine map of large order
论文作者
论文摘要
让$ g $成为一个小组。固定自动形态的$ x \ mapsto x^αg$的函数$ g \ rightarrow g $，用于固定的自动形态$ g $的$ g $ $ g $ $ g $ $ g \ g \ in g $的固定$ g \称为$ g $的仿射地图。在本文中，我们研究了有限的$ g $，其中有大量仿射地图。更确切地说，我们表明，如果$ g $允许订单的仿射图大于$ \ frac {1} {2} {2} | g | $，则$ g $最多可衍生长度，最多可衍生$ 3 $。我们还表明，更普遍地，对于\ weft（0,1 \右）$的每个$ρ\，如果$ g $接收到订单的仿射图，那么至少$ρ| g | $，那么最大的$ g $的可解决的正常亚组的长度最多，最多是$ 4 \ lfloor \ lfloor \ log_2（ρ_2（ρ^^{ - 1}}）\ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $+3 $ 3 $+3 $+3 $ 3 $+rf rf rf。
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 $4\lfloor\log_2(ρ^{-1})\rfloor+3$.
