论文标题
关于有限可溶组的较高换向器亚组的努力
On nilpotency of higher commutator subgroups of a finite soluble group
论文作者
论文摘要
让$ g $为有限的可溶性组，$ g^{（k）} $ $ k $ $ g $的$ k $第$ th。我们证明$ g^{（k）} $在且仅当$ | ab | = | = | a | a || b | $的任何$δ_k$ -values $ a，b \ in g $ in g $ in g $ in g $ in g $时。在证明过程中，我们确定了独立利息的以下结果：让$ p $为$ g $的Sylow $ p $ -subgroup。然后，$ p \ cap g^{（k）} $由$ p $中包含的$δ_k$ - 值生成。这与所谓的焦点亚组定理有关。
Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $δ_k$-values $a,b\in G$ of coprime orders. In the course of the proof we establish the following result of independent interest: Let $P$ be a Sylow $p$-subgroup of $G$. Then $P\cap G^{(k)}$ is generated by $δ_k$-values contained in $P$. This is related to the so-called Focal Subgroup Theorem.
