论文标题
有限组的某些地图的可溶性自由基和轨道
The soluble radical and orbits of certain maps on finite groups
论文作者
论文摘要
对于每个元素，有限组$ g $中的每个元素$ u $定义了$θ_U\ colon g \ to g $ to g $ by $θ_U（g）= [g^{ - u}，g] $，并设置$θ_g（u）= \ {g \ {g \ in g \midθ_u^n（g）in g \ midθ_u^n（g）= g \ hbox n} n} n} n} n} n} n} n} n} n} n。然后$θ_U$诱导$θ_g（u）$的排列;令$β_G（u）$为$ \ {1 \} $以外的轨道数。基于J.N.的工作布雷，R.A。威尔逊和第二作者，我们表明，有限群$ g $的可溶性激进分子的索引在$ 2 $ emlement $ u $的$β_g（u）$的值方面是有限的。
For each element $u$ in a finite group $G$ define a map $θ_u\colon G\to G$ by $θ_u(g)=[g^{-u},g]$ and set $Θ_G(u)=\{g\in G\mid θ_u^n(g)=g \hbox{ for some } n>0\}$. Then $θ_u$ induces a permutation of $Θ_G(u)$; let $β_G(u)$ be the number of orbits apart from $\{1\}$. Building on work of J.N. Bray, R.A. Wilson and the second author, we show that the index of the soluble radical of a finite group $G$ is bounded in terms of the values of $β_G(u)$ for $2$-elements $u$.
