论文标题
在具有广泛超溶因子的分解组的残留物上
On the residual of a factorized group with widely supersoluble factors
论文作者
论文摘要
令$ \ bbb p $为所有素数。 $ g $的一个$ g $的亚组$ h $在$ g $中称为{\ it $ \ mathbb p $ -subnormal}，如果要么$ h = g $，或者存在一系列子组$ h = h_0 \ h_1 \ le h_1 \ le h_1 \ le \ le \ le \ le \ ldots \ le h_n = g， p，\ \ foralli。$ a组$ g $称为{\ s superSoluble}，$ \ mathrm {w} $ - superSoluble，如果$ g $的每个Sylow子组为$ \ Mathbb p $ -subnormal in $ G $中。 $ g = ab $带有$ \ mathbb p $ -subnormal $ \ mathrm {w} $ - superSoluble子组$ a $ a $和$ b $。获得其$ \ mathrm {w} $ - supersoluble残差的结构。特别是，它与$ \ nilcal {a} $的nilpotent残差相吻合 - $ g $的残差。这里$ \ Mathcal {a} $是所有具有Abelian Sylow子组的组的组。此外，我们为$ \ mathrm {w} $ - 此类$ g $的超溶解度获得了新的足够条件。
Let $\Bbb P$ be the set of all primes. A subgroup $H$ of a group $G$ is called {\it $\mathbb P$-subnormal} in $G$, if either $H=G$, or there exists a chain of subgroups $H=H_0\le H_1\le \ldots \le H_n=G, \ |H_{i}:H_{i-1}|\in \Bbb P, \ \forall i.$ A group $G$ is called {\it widely supersoluble}, $\mathrm{w}$-supersoluble for short, if every Sylow subgroup of $G$ is $\mathbb P$-subnormal in $G$. A group $G=AB$ with $\mathbb P$-subnormal $\mathrm{w}$-supersoluble subgroups $A$ and $B$ is studied. The structure of its $\mathrm{w}$-supersoluble residual is obtained. In particular, it coincides with the nilpotent residual of the $\mathcal{A}$-residual of $G$. Here $\mathcal{A}$ is the formation of all groups with abelian Sylow subgroups. Besides, we obtain new sufficient conditions for the $\mathrm{w}$-supersolubility of such group $G$.
