有限群的SS-可补子群
On SS-Supplemented Subgroups of Finite Groups
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摘要: 设H是有限群G的子群.如果存在G的一个子群K，使得G=HK且H∩K在K中S-拟正规，则称H在G中SS-可补.证明了：（i）设p是整除群G阶的最小素因子.如果存在G的一个Sylow p-子群P，使得P的每个极大子群在NG（P）中SS-可补，且P'在G中S-拟正规，则G是p-幂零群.（ii）设F是一个包含超可解群类U的饱和群系，H是群G的一个正规子群，使得G/H∈F.如果对H的每一个Sylow p-子群P，P的每个极大子群在NG（P）中SS-可补，且P'在G中S-拟正规，则G∈F.Abstract: A subgroup H of a group G is said to be SS-supplemented in G if there exists a subgroup K of G such that G=HK and H∩K is S-quasinormal in K. In this paper, we show the following results:(i) Let p be the smallest prime dividing|G|and SS-supplemented in NG(P) and P' is S-qusinormal in G, then G is p-nilpotent. (ii) Let F be a saturated formation containing the class of all supersolvable groups U and let H is normal in G such thatG/H∈F. Suppose that, for all primes p dividing|H|and for all P∈Sylp(G), every maximal subgroup of P is SS-supplemented in NG(P) and P' is S-quasinormal in G. Then G∈F.
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Key words:
- SS-supplemented subgroups /
- normalizer /
- maximal subgroups /
- saturated formations .
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