On the intersection of the normalizers of derived subgroups of all subgroups of a finite group
On the intersection of the normalizers of derived subgroups of all subgroups of a finite group
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DOI：
10.1016/j.jalgebra.2009.12.015
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发表时间：
2010-03
影响因子：
0.9
通讯作者：
Shirong Li;Zhencai Shen
中科院分区：
文献类型：
--
作者：
Shirong Li;Zhencai Shen
给定一个有限群G，我们定义子群D（G）为G的所有子群的导子群的正规化子的交。设置D 0 =1。定义Di+1（G）/Di（G）=D（G/Di（G）），其中i ≠ 1。用D∞（G）表示升级数的终项.证明了导子群G′是幂零的当且仅当G=D∞（G）.若G的所有素数阶元都在D（G）中，则G是可解的，且Fitting长度至多为3.在第三节中，证明了如果群G满足G=D（G），则G′是幂零的，G″的幂零类至多为2。
Given a finite group G, we define the subgroup D(G) to be the intersection of the normalizers of derived subgroups of all subgroups of G. Set D0=1. Define Di+1(G)/Di(G)=D(G/Di(G)) for i⩾1. By D∞(G) denote the terminal term of the ascending series. It is proved that the derived subgroup G′is nilpotent if and only if G=D∞(G). Furthermore, if all elements of prime order of G are in D(G), then G is soluble with Fitting length at most 3. In Section 3, it is proved that if the group G satisfies G=D(G), then G′is nilpotent and G″has nilpotency class at most 2.