论文标题
在有限的群体中，固定点的固定点有限
On finite groups with an automorphism of prime order whose fixed points have bounded Engel sinks
论文作者
论文摘要
$ g $的元素$ g $的左engel水槽是一套$ {\ mathscr e}（g）$，以便每$ x \ in g $ in g $ in g $所有足够长的换向器$ [... [[x，g]，g]，g]，\ dots，\ dots，g] $属于$ {\ mathscr e}（g）（g）$。 （因此，$ g $是当我们可以选择$ {\ mathscr e}（g）= \ {1 \ {1 \} $时。最多$ m $，那么第二拟合子组$ f_2（g）$的索引以$ m $为限制。 $ g $的元素$ g $的右engel汇入$ {\ mathscr r}（g）$，以便每$ x \ in g $ in g $ in g $所有足够长的换向器$ [... [[g，x]，x]，x]，\ dots，x]，\ dots，x] $属于$ {\ mathscr r}（g）（g）$。 (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) We prove that if a finite group $G$ admits an automorphism $φ$ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(φ)$ has a right Engel sink of cardinality最多$ m $，然后以$ m $为符合$ m $的拟合子组$ f_1（g）$的索引。
A left Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is a left Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) We prove that if a finite group $G$ admits an automorphism $φ$ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(φ)$ has a left Engel sink of cardinality at most $m$, then the index of the second Fitting subgroup $F_2(G)$ is bounded in terms of $m$. A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) We prove that if a finite group $G$ admits an automorphism $φ$ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(φ)$ has a right Engel sink of cardinality at most $m$, then the index of the Fitting subgroup $F_1(G)$ is bounded in terms of $m$.
