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PureMathematicsnØêÆ,2020,10(1),30-37
PublishedOnlineJanuary2020inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2020.101006
OnFiniteGroupsofSSH-Subgroups
JianquanLiang
GuangxiUniversity,NanningGuangxi
Email:934819017@qq.com
Received:Dec.20
th
,2019;accepted:Jan.10
th
,2020;published:Jan.17
th
,2020
Abstract
LetGbeafinitegroup.AsubgroupHofGiss-permutableinGifHpermuteswith
everySylowsubgroupofG.AsubgroupHofGiscalledanSSH-subgroupinGifG
hasans-permutablesubgroupKsuchthatH
sG
=HKandH
g
∩N
K
(H)≤H,forall
g∈G,whereH
sG
istheintersectionofalls-permutablesubgroupsofGcontainingH.
ThisarticlestudiesthestructureoffinitegroupswithSSH-subgroupwhichisprime
power order.Some characterizations of a finite group as a p-nilpotent group are given.
Keywords
SSH-Subgroups,p-NilpotentGroups,Sylowp-Subgroups
,f+•SSH-f+k•+
ùùùjjj
2܌Ƨ2ÜHw
Email:934819017@qq.com
ÂvFϵ2019c1220F¶¹^Fϵ2020c110F¶uÙFϵ2020c117F
©ÙÚ^:ùj.,f+•SSH-f+k•+[J].nØêÆ,2020,10(1):30-37.
DOI:10.12677/pm.2020.101006
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Á‡
H´+G˜‡f+§XJ•3G˜‡s-˜†f+K§¦H
sG
= HK¿…é?¿g∈G
ÑkH
g
∩N
K
(H)≤H¤á§K¡H•GSSH-f+"Ù¥H
sG
´G•¹XH•
s-˜†f+"©ÙïÄäkƒê˜SSH-f+k•+(§‰Ñk•+•p-˜"+˜
•x^‡"
'…c
SSH-f+§p-˜"+§Sylowp-f+
Copyright
c
2020byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
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g§=H´+G˜‡f+§•3G˜‡5f+K¦H
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DOI:10.12677/pm.2020.10100631nØêÆ
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g
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(H) ≤HѤá§K¡H•GSSH-f+§Ù¥H
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´G•¹X
H•s-˜†f+"
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¼±eA‡¤J"
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+…=Pz‡4Œf+Ñ´GSSH-f+"
½n1.2-N´G˜‡5f+§¦G/N´‡Œ)+§eNšÌ‚Sylowp-f+
4Œf+´GSSH-f+§KG´‡Œ)+"
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1)eH≤K§KH•´KSSH-f+¶
2)eN•G5f+§…N≤H§@oH´GSSH-f+…=H/N´G/N
SSH-f+¶
3) eH´G˜‡p-f+§N´G˜‡p
0
-f+§@oHN´G/NSSH-f+§…
HN/N´G/NSSH-f+"
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Ún2.3 ([6]§Ún2.2)HÚK´Gf+§…H≤K§H´Gs-˜†f+§KH
sG
´Gs-˜†f+…H
sG
≤H
G
"
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O
p
(G) ≤N
G
(H)"
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J|P|≤p
2
§…G†A
4
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"+…=N
G
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G
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p
0
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1 <|D|<|P|§¦P¥|D|½4(e|D|= 2 )f+•GH-f+§KP≤Z
U
(G)"
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+"KG´p-˜"+…=Pz‡p½4(ep= 2)Ì‚f+´GSSH-f+"
DOI:10.12677/pm.2020.10100632nØêÆ
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P≤Z
U
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U
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d([12]§½n1.7.19)kP≤Z
U
(G)"eΦ(P)=1§KP´†+"N•G45f+
…N≤P§KG/N÷vb^‡§P/N≤Z
U
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1
§K
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1
≤Z
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1
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/N
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1
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˜‡SSH-f+§K•3G˜‡s-˜†f+K§¦L
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SSH-f+§KP≤Z
U
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ÚÚn2.14§M´p-˜"+§ÏdM= P×Q§|G/C
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(P)|´2•˜§¿…d( [12]§Appendix
C½n6.3)kP≤Z
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ep>2§N´G45f+§-H´Ppf+"db^‡§H´GSSH-
f+§=•3G˜‡s-˜†f+K§¦H
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g
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Ún2.14-P´G˜‡š²…5p-f+§eP¥•3f+D§÷v1 <|D|<|P|§¦
P¥|D|½4(e|D|= 2 )f+•GSSH-f+§KP≤Z
U
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Z
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(G)"b•3P÷v|H|=|D|f+H§¦HØ´GH-f+§db^‡§
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g
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K
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P∩M´P4Œf+"e|P:D|=p§KdÚn2.11P≤Z
U
(G)"b|P:D|>p§K
P∩Mz‡|D|½4(e|D|=2 )f+Ñ´G˜‡SSH-f+"d8Bb§P∩M≤
Z
U
(G)§Ï•|P/P∩M|= p§ŒíÑP≤Z
U
(G)"
3.̇(J
½n3.1p´+G•ƒÏf"P´G˜‡Sylowp-f+§XJG†A
4
Ã'§
…P¥p
2
f+•GSSH-f+§KG•p-˜"+"
y²b½nØý§¿G´•4‡~§KdÚn2.5•§|P|≥p
3
"M<G…
p||M|§M
p
•M˜‡Sylowf+"e|M
p
|p
2
§dÚn2.5§M•p-˜"+"e|M
p
|>p
2
§d
DOI:10.12677/pm.2020.10100633nØêÆ
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8Bb9Ún2.1(1)§M•p-˜"+"Œ±bG´4šp-˜"+§d([13],Satz5.4,
p.434)G= PQ§Ù¥Q•Ì‚+§ExpP=p½ExpP≤4(p=2ž)§P/Φ(P) ´G
ÌÏf§Ù¥Q∈Syl
q
(G)…q6= p"
3P/Φ(P)¥˜‡ƒx§K|hxi|= p½|hxi|= 4"
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hxi
sG
=hxiK§…hxi
g
∩N
K
(hxi)≤hxi,∀g∈G"K=Gž§hxi•GH-f+§…hxiG§
dÚn2.2§hxiG§KkhxiΦ(P)/Φ(P)G/Φ(P)§qÏ•P/Φ(P)´G/Φ(P)ÌÏf§Ïk
hxiΦ(P) = P§KP= hxiΦ(P) = hxi§dÚn2.6•§G´˜"+§gñ"
|hxi|=p§duPØ´Ì‚+§ŒP¥ƒa¦a/∈hxi§-B=hai×hxi§
KB´˜‡p
2
f+"db^‡§B´G˜‡SSH-f+§•3˜‡s-˜†f+K§
¦B
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½BΦ(P)=P"eBΦ(P)=Φ(P)§KB≤Φ(P)§†hxiΦ(P)§gñ"eBΦ(P)=P§K
P=BΦ(P)=B=hai×hxi§†|P|≥p
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p
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p
(G)G§ÏdQG§gñ"ÏdO
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≤H
G
≤P§…H
sG
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sG
P§K
H
sG
QG…dG45ŒíѧH
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G
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KK= P§gñ"ÏdKNP§w,KN/NG/N§…dG45§P/N´G/N45f
+§KN/N=1"ù¿›XK≤N"ÏdP=HN=H´Ì‚+"G´p-˜"+§gñ"
l½ny"
½n3.2p´+GÛƒÏf"bL´G˜‡5f+§¦G/L´p-˜"+…
P´LSylowp-f+"e•3Pf+D§÷v1<|D|<|P|§¦Pz‡•|H|=|D|
f+H´GSSH-f+§…N
G
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y²b½nØý§¿G•4‡~"
(1)O
p
0
(G) = 1"
PT=O
p
0
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∼
=
G/LT´p-˜"+"-
HT/T´PT/T|D|f+§Ù¥§H´P|D|f+§Ï•H´GSSH-f+§dÚ
n2.1(3)kHT/T´G/TSSH-f+"2ÏN
G
(P)´p-˜"+§N
G/T
(PT/T) = N
G
(P)T/T´
p-˜"+"G/T÷vb^‡§dG45•§G/T´p-˜"+§gñ"
(2)-K•G˜‡ýf+§¦P≤K§KK´p-˜"+"
d Ún2.1(1)•§Pz‡|D|f+H´KSSH-f+§ÏN
K
(P)≤N
G
(P) …N
G
(P)
´p-˜"+§N
K
(P) ´p-˜"+"K÷vb^‡§dG45•K´p-˜"+"
(3)L= G"
eL<G§Kd(2)•L´p-˜"+"-T•L5p-Ö§KTcharLG§TG…Š
DOI:10.12677/pm.2020.10100634nØêÆ
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â(1)kT= 1§?íÑL= P…G= N
G
(P) ´p-˜"+§gñ"
(4)O
p
(G) 6= 1"
•Ä+Z(J(P))§Ù¥J(P)´PThompsonf+§eN
G
(Z(J(P)))<G§d(2)
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ë•©z
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