论文标题
$ \ Mathcal {x} $ - $ P $ - 组的系列和Abelian子组的补充
The $\mathcal{X}$-series of a $p$-group and complements of abelian subgroups
论文作者
论文摘要
令$ g $为$ p $ -group。我们用$ \ Mathcal {x} _i（g）$ $ g $的所有子组的交集，$ g $，具有索引$ p^i $，对于$ i \ leq \ log_p（| g |）$。在本文中，研究了新引入的系列$ \ {\ Mathcal {x} _i（g）\} _ I $，并证明了有关其行为的许多结果。作为这些结果的应用，我们表明，如果Abelian子组$ a $ a $ g $相交的每个子组$ \ Mathcal {x} _i（g）$ at $ \ MATHCAL {x} _i _i（a）$，那么$ a $ a $在$ g $中具有补充。相反，如果任意子组的$ g $的$ h $具有正常的补充，则$ \ mathcal {x} _i（h）= \ Mathcal {x} _i（g）\ cap h $。
Let $G$ be a $p$-group. We denote by $\mathcal{X}_i(G)$ the intersection of all subgroups of $G$ having index $p^i$, for $i \leq \log_p(|G|)$. In this paper, the newly introduced series $\{\mathcal{X}_i(G)\}_i$ is investigated and a number of results concerning its behaviour are proved. As an application of these results, we show that if an abelian subgroup $A$ of $G$ intersects each one of the subgroups $\mathcal{X}_i(G)$ at $\mathcal{X}_i(A)$, then $A$ has a complement in $G$. Conversely if an arbitrary subgroup $H$ of $G$ has a normal complement, then $\mathcal{X}_i(H) = \mathcal{X}_i(G) \cap H$.
