a generalization of $c$-supplementation

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1、Ageneralizationofc-Supplementation∗ShihengLi,DengfengLiang,WujieShiSchoolofMathematicScience,SuzhouUniversity,Suzhou215006,ChinaE-mail:lishiheng01@163.com;dengfengliang@163.com;wjshi@suda.edu.cnAbstractAsubgroupHissaidtobenc-supplementedinagroupGifthereisa

2、subgroupK≤GsuchthatHKGandH∩KiscontainedinHG,thecoreofHinG.WecharacterizethesolvabilityoffinitegroupsGwithsomesubgroupsofSylowsubgroupsnc-supplementedinG.Wealsogivearesultonc-supplementedsubgroups.Keywordssolvable,nc-supplemented,p-nilpotent.Wang[6]introduc

3、esthenotationofc-supplementedsubgroupsanddeterminesthestruc-tureoffinitegroupsGwithsomesubgroupsofSylowsubgroupsc-supplementedinG.Herewegiveanewconceptcallednc-supplementationthatisaweakversionofc-supplementationandcharacterizethesolvabilityofgroupsGwithsom

4、emaximalor2-maximalsubgroupsnc-supplementedinG,respectively.Inthispaper,πdenotesasetofprimes.WesayG∈EπifGhasaHallπ-subgroup;G∈CπifG∈EπandanytwoHallπ-subgroupsofGareconjugateinG;G∈DπifG∈Cπandeveryπ-subgroupofGiscontainedinaHallπ-subgroupsofG.Wesaythatanumbe

5、rnisaπ-numberifeveryofitsprimedivisorisinπ.

6、G

7、πdenotesthelargestπ-numberthatdivides

8、G

9、.H<·GdenotesthatHisamaximalsubgroupofG.Liscalleda2-maximalsubgroupofG,ifthereexistsamaximalsubgroupMofGsuchthatL<·M.Definition1.LetGbeagroupandHbeasubgroupofG.(1)Hissaidto

10、bec-supplementedinGifthereexistsasubgroupKofGsuchthatTHK=GandH∩K≤HG,whereHG=HgisthecoreofHinG.WesaythatKisag∈GarXiv:math/0509447v1[math.GR]20Sep2005c-supplementofHinG.(2)Hissaidtobenc-supplementedinGifthereisasubgroupKofGsuchthatHKGandH∩K≤HG.WesaythatKisa

11、nc-supplementofHinG.2Remark1.LetHbenc-supplementedinG.Evidently,Hisc-supplementedinGifGissimple;Hisc-supplementedinGifHisamaximalsubgroupofG.2Ingeneral,nc-supplementationdoesnotimpliesc-supplementation.Example.LetG=A4andB={(1),(12)(34),(13)(24),(14)(23)}.S

12、etC={(1),(12)(34)}andD={(1),(14)(23)}.ThenC×D=BGandCisnc-supplementedinG.However,Cisnotc-supplementedsinceCG=1andA4hasnosubgroupoforder6.2Lemma1.IfHisnc-supplementedinG,thenthereexistsasubgroupCofGsuchthatH∩