摘要
对所有子群或为类正规或为自正规的有限群(称为PS群)进行了研究,获得了这类群的一些性质,并在极大子群为幂零或内幂零的条件下获得了这类群的分类.主要结果为:设G是一个PS群,则G的极大子群为幂零或内幂零当且仅当G为下列群之一:(1) G是Dedekind群;(2 ) G =〈a,b|ap =bqn =1,ab =aλ,q|p - 1,p|λq - 1,p |/λ- 1〉;(3) G =〈a,b,c|ap =bqβ=cr =1,[b,c]=[a,c]=1,ab =aλ,p|/λ- 1,p|λq - 1,q|p - 1,r|λ- 1〉;(4)G =〈a,b,c|ap =bq =cr =1,[b,c]=1,ab =aλ,ac =as,p |/ (λ- 1) (s- 1) ,p |λq - 1,p |sr - 1,rq|p - 1〉,q >r;(5 ) G =〈a,c|aq =crn =1,ac =aλ,q|/λr - 1,q|λr2 - 1,r2 |q - 1〉,n≥2 ;(6 ) G =〈a,c|aq2 =crn =1,ac =aλ,q|/λ- 1,q2 |λr - 1,r|q - 1〉;(7) G =〈a,b,c|aq =br =crn- 1 =1,[a,b]=[b,c]=1,ac =aλ,q|/λ- 1,q|λr - 1,r|q- 1〉,n≥2 ;(8) G =〈a,b,c|aq =b4 =c4 =1,b2 =c2 ,[a,b]=1,ac =a- 1 ,bc =b- 1〉,q是奇素数;(9) G =〈a,b,c|ap =bq =crn =1,[a,b]=1,ac=aλ,bc=bλ,p |/λ- 1,q|/λ- 1,pq|λr - 1,r|p - 1,r|q - 1〉.
The finite groups whose subgroups are pronormal or selfnormalizing are studied, while some properties and classifications are obtained. The main result is the following: Suppose G is a PS group, then all the maximal subgroups of G are either nilpotent or inner nilpotent if and only if G is one of the following groups: (1) G is a Dedekind group; (2) G=〈a,b|a p=b q n =1,a b=a λ,q|p-1,p|λ q-1, p｜/ λ-1〉;(3) G=〈a,b,c|a p=b q β =c r=1,[b,c]=[a,c]=1,a b=a λ, p｜/ λ-1,p|λ q-1,q|p-1, r|λ-1〉;(4) G=〈a,b,c|a p=b q=c r=1,[b,c]=1,a b=a λ,a c=a s, p｜/ (λ-1)(s-1),p|λ q-1,p|s r-1,rq|p-1〉,q>r;(5) G=〈a,c|a q=c r n =1,a c=a λ, q｜/ λ r-1,q|λ r 2 -1,r 2|q-1〉,n≥2;(6) G=〈a,c|a q 2 =c r n =1, a c=a λ, q｜/ λ-1,q 2|λ r-1,r|q-1〉;(7) G=〈a,b,c|a q=b r=c r n-1 =1,[a,b]=[b,c]=1,a c=a λ, q｜/ λ-1,q|λ r-1,r|q-1〉,n≥2;(8) G=〈a,b,c|a q=b 4=c 4=1,b 2=c 2,[a,b]=1,a c=a -1 ,b c=b -1 〉,(q odd prime);(9) G=〈a,b,c|a p=b q=c r n =1,[a,b]=1,a c=a λ,b c=b λ, p｜/ λ-1, q｜/ λ-1,pq|λ r-1,r|p-1,r|q-1〉.
出处
《浙江大学学报（理学版）》
CAS
CSCD
北大核心
2005年第3期241-245,共5页
Journal of Zhejiang University（Science Edition）
基金
国家自然科学基金资助项目 (10 1710 89)
中国民航学院博士基金资助项目
关键词
类正规
自正规
极大子群
pronormal
selfnormalizing
maximal subgroup
