关于半正规的几类群

利用子群的半正规性讨论了几类有限群的结构,得到如下主要结果:(l)极大子群超可解的有限群当其极大子群的极小子群半正规时,它不是超可解群就是如下三种群之一:(I)p~αq~β阶内-Abel群,p(?)q-1;(Ⅱ)p~(α+β)r(?)阶群,α≥2,β≥0,p~β=│φ(G)│,p~(α-1)||r—1,α~((?)~α+β)=c_1~(?)=c_2~(?)=…=c_(?)~(?)=1,c_ic_j=c_jc_i,i,j=1,2,…,p,c_(?)~(?)=c_(i+1),i=1,2,…,p-1,c_(?)~(?)=c_1~(?),t(mod r)指数p~(α-1);(Ⅲ)D_(2_q)型群;(2)极大子群可解的非Abel有限单群当其二次极大子群的极小子群半正规时,G恰为A_5.

SEVERAL TYPES OF GROUPS THE SEMI-NORMALITY

A subgroup H of a finite group G is called to be semi-normal if theve exists a subgroup B such that G = HB and for any proper subgroup B, of B, HB1 is a proper subgroup of G. The following results were provrn with the concept :( 1 ) Let G be a finite group. If any maximal subgroup M of G is supersolvable and any minimal subgroups of M are semi-normal in M , then G is supersolvable, or is the following one of the three types :( I ) A inner- Abelian group of order( II ) A gruop of order( III ) A group of type of D2(2) A non-Abelian finite simple group G is a alternating group of degree 5 if any maximal subgroups of G are solvable and minimal subgroups of any 2-maximal subgroups M of G are semi-normal in M.