通过p-元可解的radical成员判别法
Criteria for solvable radical membership via p-elements
摘要：
Guralnick、Kunyavskii、Plotkin和Shalev证明了有限群$G$的可解根可以刻画为所有$x\in G$的集合，使得$<x,y>$对所有$y\in G$都可解。我们证明了这个结果的两个推广。首先，对于每个$p$ -元$y\in G$，对于每个奇素数$p$，检查$<x,y>$的可解性就足够了。其次，如果$x$是奇数阶的，那么对每个2元$y\in G$检验$<x,y>$的可解性就足够了。
英文摘要：
Guralnick, Kunyavskii, Plotkin and Shalev have shown that the solvable radical of a finite group $G$ can be characterized as the set of all $x\in G$ such that $<x,y>$ is solvable for all $y\in G$. We prove two generalizations of this result. Firstly, it is enough to check the solvability of $<x,y>$ for every $p$-element $y\in G$ for every odd prime $p$. Secondly, if $x$ has odd order, then it is enough to check the solvability of $<x,y>$ for every 2-element $y\in G$.
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