H-子群的norm The Norm of H-Subgroups

doi:10.12677/PM.2019.97104

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PureMathematicsnØêÆ,2019,9(7),799-803

PublishedOnlineSeptember2019inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2019.97104

TheNormofH-Subgroups

HongfangGu,L¨uGong

∗

SchoolofSciences,NantongUniversity,NantongJiangsu

Received:Aug.2

,2019;accepted:Aug.28

,2019;published:Sep.4

,2019

Abstract

Inordertoinvestigatethestructureofﬁnitenilpotentgroup,anewequivalentchar-

acterizationofﬁnitemeta-nilipotentgroupisobtainedbythenormofH-subgroups.

Keywords

Norm,SolubleGroup,SubnormalSubgroup,H-Subgroup

H-f+norm

ùùù•••§§§÷÷÷ÆÆÆ

∗

HÏŒÆnÆ§ô€HÏ

ÂvFÏµ2019c82F¶¹^FÏµ2019c828F¶uÙFÏµ2019c94F

Á‡

•?˜Ú&¢k•˜"+(§|^H- f+norm§‰Ñk•æ˜"+˜‡#d

∗ÏÕŠö"

©ÙÚ^:ù•,÷Æ.H-f+norm[J].nØêÆ,2019,9(7):799-803.

DOI:10.12677/pm.2019.97104

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•x"

'…c

Norm§Œ)+§g5f+§H-f+

2019byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

1.Só

©ïÄ+Ñ´k•+"©Ì‡8´|^+H-f+‰Ñ+‡¥%#A"

H-f+Äk3©[1]¥JÑ§+G¥f+H÷véz‡g∈GÑkN

(H) ∩H

≤H§K

¡H´GH-f+"-H(G)L«+G¤kH-f+8Ü§=

H(G) = {H≤G|N

(H)∩H

≤H,∀g∈G}.

|^H-f+§©[1]‰Ñz‡g5f+Ñ5k•Œ)+A§¿•xz‡f+½

5½g5k•+("'uH-f+?˜ÚïÄŒ±ëw©[2–5]"

+GnormN(G)´•+G¥¤kf+5zf§ù˜VgÄkdBaer3©[6]¥J

Ñ",˜‡†ƒƒ'f+´Wielandtf+§ù´dWielandt3©[7]¥0+¥¤kg5

f+5zf"WielandtgŽÌ‡'5g5f +,Ø´¤kf +"©·‚òYù

g´§•Ä+¥H-f+†+enorm§=+¥¤kH-f+5zf"

·‚ÎÒÚâŠÑ´IO§PL«+GSylowp-f+§G

L«+G+§Z

∞

(G)L

«+G‡¥%§Ù¦ÎÒŒëw©[8–12]"

Äk·‚‰ÑXe½Â

½Â1.1.-G´˜+"KA

(G)L«H(G)†G

¥z‡f+5zf§=

(G) =

H∈H(G),H≤G

(H),

w,§A

(G) EG"eG

˜"§KG

¥z‡f+H3G¥Ñg5§u´dH∈H(G)

•H3G¥5§ÏdG=A

(G)"Ïd·‚g,Ž•Ù_ ·K´Ä(?e©·‚ò‰Ñ

DOI:10.12677/pm.2019.97104800nØêÆ

ù•§÷Æ

’½‰Y"

éuù‡½Â§·‚‰XeA‡`²µ

(1)

H∈H(G),H≤G

(H)†

H∈H(G

)

(H)´ØÓf+"

d+G¥H(G)ƒ'5Ÿ•µeH∈H(G),H≤G

§KH∈H(G

)"ù‡5Ÿ‡L5¿

Ø(§Ïdþã½ÂØÓ"

(2)

H∈H(G),H≤G

(H)†

H∈H(G)

)´ØÓf+"

¦+üöÑ´ïÄ+G¥H(G)f+§5zfé–´H†H

§g,Ø˜"

2.Ì‡Ún

Äk·‚0˜'u+¥H-f+ÄÚnÚ5Ÿ"

Ún2.1.-G´˜‡+"

(1)[1]Lemma2(1)eN≤H…NEG§KH∈H(G)…=H/N∈H(G/N)"

(2)[1]Theorem6(2)eH∈H(G)…HEEK≤G§KHEK"

(3)[1]Theorem6(3)eH∈H(G)§NEG…N≤N

(H)§KN

(HN)=N

(H)…HN∈

H(G)"

(4)[1]Proposition3(2)eN,KEG…K≤N§KN/KSylowf+S/K3G_–Sá

uH(G)"AO/§G5f+Sylowf+ÑáuH(G)"

Ún2.2.[13]Theorem6.3-G´˜‡+§p´˜‡ƒê"eP´G¥¦G/C

(P)´p•

˜5p-f+§KP≤Z

∞

(G)"

e5§·‚‰Ñ+GA

(G)˜k^5Ÿ"

5Ÿ2.3.-G´˜+"eNEG…N≤A

(G)∩G

§KA

(G/N) = A

(G)/N"

y².-NEG…N≤A

(G)∩G

"·‚•ÄG/Nf+H/N "

(G/N)=

H/N∈H(G/N),H/N≤(G/N)

G/N

(H/N)

H∈H(G),N≤H≤G

G/N

(H/N),d[Ún2.1(1)],N≤G

H∈H(G),N≤H≤G

(H)/N

dÚn2.1(3)ÚN≤A

(G)•µ

(G)=

H∈H(G),H≤G

(H)

H∈H(G),H≤G

(HN)

H∈H(G),N≤H≤G

(H)

Ïd§·‚kA

(G)/N= A

(G/N)"

DOI:10.12677/pm.2019.97104801nØêÆ

ù•§÷Æ

3.Ì‡½n

©[14]¥y²éu?¿+GÑkN(G) = 1…=Z(G) = 1"g,/§·‚Ž‡•X

eØä´Ä•(µ

éu?¿+GÑkA

(G) = 1…=C

) = 1?

Xe½n‰Ñþã¯KÈ4‰Y"

½n3.1.-G´+"K

(1)C

) ≤A

(G)"

(2)A

(G) = G…=G

˜""

(3)C

) = 1…=A

(G) = 1"

y².(1)w,§·‚k

)=∩

H≤G

(H)

≤∩

H≤G

,H∈H(G)

(H)

≤∩

H≤G

,H∈H(G)

(H) = A

(G)

(2)eG

˜"§KG

¥z‡f+H3G¥g5"dÚn2.1(2)•eH∈H(G)§KH

3G¥5§=G= A

(G)"

‡L5§bG= A

(G)"-T≤G

´GÜ¤Ïf…L´T˜‡Sylowf+"u´d

Ún2.1(2)•L∈H(G)§L3G¥5"ÏdL= T§u´TŒ)§=A

(G)Œ)"

-H´G

˜‡Sylowf+"dÚn2.1(2)•H•¹3H(G)¥"u´dG= A

(G)•H

3G¥5"ÏdH3G

¥5…G

˜""

(3)w,§d5Ÿ3.3(1)•µeA

(G) = 1§KC

) = 1"

‡L5§eC

) = 1§KZ(G

) = 1"-M= A

(G)∩G

"·‚ÄkbM6= 1"Ï•G

´Œ)+§¤±3M¥•3G45f+N§Ø”N´Ð†p-f+"

-Q´G

Sylow q-f+§Ù¥p6= q"u´Q≤C

(N)…G

(N)´p•˜"Ïd

dÚn2.2•N≤Z

∞

) =Z(G) =1§gñ"ùÒL²M=1§…[A

(G),G

] ≤M= 1§k

) = A

(G)"

lC

) = A

(G) = 1"=§A

(G) = 1…=C

) = 1"

ë•©z

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ù•§÷Æ

https://doi.org/10.1515/jgth.2000.012

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2.ÏL•Ä•http://cnki.net/ºÜ/Î‡\•0?\•Î‡µhttp://www.cnki.net/old/§†ýÀJ/I

Ývž:Âµhttp://www.hanspub.org/Submission.aspx

Ïre‡µpm@hanspub.org

DOI:10.12677/pm.2019.97104803nØêÆ