论文标题
关于某些有限组的某些直接产物中奇数的亚组的正态
On the Pronormality of Subgroups of Odd Index in some Direct Products of Finite Groups
论文作者
论文摘要
如果$ h $，则$ g $的一个$ g $的亚组$ h $在$ h $和$ h^g $中为{\ it pronormal}，在$ \ langle h，h^g \ rangle $ in G $中的$ \ langle h，h^g \ rangle $。有限群体理论，组合和置换群体理论中的某些问题是用正态来解决的，因此，特定子组在给定群体中的正态性问题是感兴趣的。有限组中奇数指数的亚组满足了天然的正态条件。在本文中，我们继续研究奇数指数亚组的正态性，并考虑一些有限基团的直接产物中奇数索引亚组的正态问题。 特别是，在本文中，我们证明，奇数索引的亚组在直接乘积$ g $的$ g $中是奇数特征的字段上的$ g $，并且仅当奇数索引的亚组在$ g $的每个直接因子中都是正态的。此外，确定简单符号群体在奇数特征领域的直接乘积中给定的奇数子组的正态可靠，以确定$ \ prod_ = 1}^t \ nath sym syms y = 1}^t \ sym syms of $ \ prod_ = 1}^t \ wr wr wr prop_ = 1}^w wer__ = n_ n_ n_ n_ n_ nory of $ h $ h $奇数索引的正态。 $ sym_ {n_i} $在$ \ {1，\ dots，n_i \} $上自然作用，因此$ h $ projects $ \ prod_ {i = 1}^t sym_ {n_i} $。因此，在本文中，我们获得了$ \ prod_ {i = 1}^t \ mathbb {z} _ {p_i} \ wr sym_ {n_i {n_i} $的子组中的子组$ h $奇数索引的正态标准$ \ {1，\ dots，n_i \} $，因此$ h $ projects $ \ prod_ {i = 1}^t sym_ {n_i} $。
A subgroup $H$ of a group $G$ is said to be {\it pronormal} in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for each $g \in G$. Some problems in Finite Group Theory, Combinatorics, and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper we prove that the subgroups of odd index are pronormal in the direct product $G$ of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of $G$. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup $H$ of odd index in a subgroup of $\prod_{i=1}^t \mathbb{Z}_3\wr Sym_{n_i}$, where each $Sym_{n_i}$ acts naturally on $\{1,\dots, n_i\}$, such that $H$ projects onto $\prod_{i=1}^t Sym_{n_i}$. Thus, in this paper we obtain a criterion of pronormality of a subgroup $H$ of odd index in a subgroup of $\prod_{i=1}^t \mathbb{Z}_{p_i}\wr Sym_{n_i}$, where each $p_i$ is a prime and each $Sym_{n_i}$ acts naturally on $\{1,\dots, n_i\}$, such that $H$ projects onto $\prod_{i=1}^t Sym_{n_i}$.
