关于Camina群的基本定理

On the Basic Theorem of Camina Groups

设K是有限群G的一个非平凡正规子群 ,如果对于每个x∈G-K ,x与xy共轭 , y∈K ,那么 ,称G为以K为核的Camina群 .A .R .Camina建立了关于Camina群的一个基本定理 .作者给出了这个基本定理的一个初等证明 ,这个初等证明不但避免使用M .Suzuki的质幂元单群的分类定理 ,还同时改进了这个基本定理的结论 .此外 ,所用的证明方法还为基本定理的原证明过程中的一个重要引理提供了一个很简洁的证明 .最后 ,得到Camina群是Frobenius群的一个充分条件以及Camina群是以其换位子群为核的Frobenius群的一个充分条件 .

Let K be a nontrivial normal subgroup of a finite group G,the group G is called a Camina group with the kernel K if for each x∈ G-K,x is conjugate to xy,y∈K. A. R. Camina established a basic theorem on Camina groups. The purpose of this paper is to give an elementary proof for the basic theorem. This proof both avoids to use the Suzuki's theorem of the classification of simple groups in which the order of every element is a power of a prime and improves the conclusion of the basic theorem. In addition,the method provides a simple proof for a lemma which is an important step to prove the basic theorem. Finally,the author obtains the following two corollaries:(1) Let G is a Camina group with the kernel K, if all Sylow subgroups of G are abelian,then G is a Frobenius group with the kernel K. (2) Let G be a Camina group with the kernel K,if all Sylow subgroups of G are abelian and G is meta-abelian,then K=G′,the commutator subgroup of G,and G is a Frobenius group with the kernel G′.