Sylow p-子群为循环群的10pn阶非交换群的 Coleman自同构群 On Coleman Outer Automorphism Groups of Non-Commutative Groups of Order 10pn with Sylow p-Subgroup Being Cyclic Groups

doi:10.12677/PM.2021.118165

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PureMathematicsnØêÆ,2021,11(8),1475-1481

PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.118165

Sylowp-f+•Ì‚+10p

š†+

ColemangÓ+

•••»»»CCCKKK

∗

§§§°°°???‰‰‰

†

“ŒÆêÆ†ÚOÆ§ìÀ“

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Á‡

3ùŸ©Ù¥§|^+K4•5Ÿ‰Ñ˜aSylowp-f+•Ì‚+10p

š†+

ColemangÓ+("

'…c

K4•§ColemangÓ+§SgÓ+

OnColemanOuterAutomorphism

GroupsofNon-CommutativeGroups

ofOrder10p

withSylowp-Subgroup

BeingCyclicGroups

A’gaYihuo

∗

,JinkeHai

†

CollegeofMathematicsandStatistics,QingdaoUniversity,QingdaoShandong

∗1˜Šö"

†ÏÕŠö"

©ÙÚ^:•»CK,°?‰.Sylowp-f+•Ì‚+10p

š†+ColemangÓ+[J].nØêÆ,2021,

11(8):1475-1481.DOI:10.12677/pm.2021.118165

•»CK§°?‰

Received:Jun.30

,2021;accepted:Aug.3

,2021;published:Aug.10

,2021

Abstract

Inthisnote,weusetheprojectionlimitpropertyofgroupstogivethestructureof

Colemanautomorphismgroupsforaclassofnon-commutativegroupsoforder10p

withSylowp-subgroupbeingcyclicgroups.

Keywords

ProjectionLimit,ColemanAutomorphism,InnerAutomorphism

2021byauthor(s)andHansPublishersInc.

ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).

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

1.Úó

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¡k•+G˜‡gÓσ•G˜‡ColemangÓ§XJσ3G?¿˜‡Sylowf+þ

•›uG,‡SgÓ3Ùþ•›§@o¤kùgÓ¤Aut(G)˜‡f

+§P•Aut

Col

(G)"w,Inn(G)EAut

Col

(G)§-Out

Col

(G)=Aut

Col

(G)/Inn(G),¡Out

Col

(G)

•GColemangÓ+"

3©z[1]¥§E.C.Dadey²Out

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z[3]¥§A.V.AntwerpenïÄäkg¥%zAü5f +k•+ColemangÓ§

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äkg¥%z5f+k•+ColemangÓ§•‰ÑOut

Col

(G)=1˜¿©^‡"

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aSylowp-f+•Ì‚+10p

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¥‰Ñ+ColemangÓ+("·‚y²XeÌ‡(Jµ

DOI:10.12677/pm.2021.1181651476nØêÆ

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½n1G=<a>o§Ù¥a

−1

§…a

=1=b

§p´Œu5ƒê§

KAut

Col

(G)=Inn(G)"

½n2G=<a,b>§Ù ¥a

§…a

=1=b

,r6≡1(modp

)…r

≡1(modp

),p≡

1(mod5)§p´Œu5ƒê§KAut

Col

(G)=Inn(G)"

½n3G=<a,b,c>§Ù¥a

−1

§…a

=1=[a,c]=[b,c]§p´Œu5ƒ

ê§KAut

Col

(G)=Inn(G)"

©¥§GL«k•+§π(G)L«|G|ƒÏf8Ü§^ζ(G)L«G¥%§C

L«n

‡C

†È§C

L«p

Ì‚+§O

(G)L«G•Œ5p

−f+"Ù§PÒ´IO§ë„©

z[2,8–10]"

2.ý•£

½Â1G´k•+§N

,···,N

•G5f+"PG

=G/N

§-π

→G

•g,N§Ù¥1≤i,j≤s,s•ê§K¡

G=Projlim

1≤i≤s

,π

)={(g

,···,g

)∈

i=1

,1≤i,j≤s}.

•+G'uÓπ

K4•§{¡

G•+GK4•"

d½Â1á=íÑXeÚn1ÚÚn2µ

Ún1PÒXþ,K

G´G

×G

×···×G

f+§…G

Ún2P

GG

þÝ•π

§XJπ

•÷Ó§K

G=Projlim

1≤i≤s

,π

)={(g

,···,g

)∈

i=1

dž§P

G=Projlim

1≤i≤s

)§=Projlim

1≤i≤s

)=G

×G

×···×G

Ún3[5]G´k•+§N

,···,N

•G5f+"-π

:G/N

→G/N

•g,

N§÷v∩

i=1

=1§…é?¿p∈π(G)§–•3˜‡•Ii(1≤i≤s)¦(p,|N

|)=1§

∼

G=Projlim

1≤i,j≤s

,π

Ún4[8]G•k•Œ)+§PG

=G/O

(G),G

=G/O

(G)O

(G)§Ù¥π(G)=

,···,p

}"-π

→G

•g,N§…π

pg,Óπ

∗

/ζ(G

)→G

/ζ(G

)§

KAut

Col

(G)

∼

Projlim

1≤i,j≤n

/ζ(G

),π

∗

)

∼

Projlim

1≤i≤n

(Inn(G

))"

Ún5[7]G=<a>o§Ù¥a

−1

§…a

=1=b

§p´Œu5ƒê§

K<a>∩=1"

Ún6G=<a>o§Ù¥a

−1

§…a

=1=b

,p´Œu5ƒ ê§K{b

|i=

2,4,6,8}•G¥¤k5…•G¥%"

DOI:10.12677/pm.2021.1181651477nØêÆ

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y²o(b

)=5§Ù¥0≤i<10,0≤j<p

"i•Ûêž,da

−1

i−1

i+1

−j

i−1

i+1

−j

i−2

i+2

−j

i−2

=......=b

;

i−1

3i+1

−j

i−1

=......=b

;

KŒ

=1.

dÚn5iÃ)¶i•óêž§d(b

)

=1(Óni•Ûêy²•{)9Ún5

i=2,4,6,8,j=0.

Ïd{b

|i=2,4,6,8}•G¥¤k5"

?ab

∈G,da

−1

=abb=ba

−1

b=ba

−1

b=ba

−2

=ba

−2

−1

=ba

−3

aba

−1

=ba

−3

−2

=......=b

1−p

Ï•a

=1§¤±a

1−p

=aa

−p

=a"ab

a§Ïdb

•G¥%§=ζ(G)=C

3.Ì‡½ny²

½n1G=<a>o§Ù¥a

−1

§…a

=1=b

§p´Œu5ƒê§K

Aut

Col

(G)=Inn(G)"

y²Ï•|π(G)|=3§dÚn6§ζ(G)=C

§G=<a>o=(C

)×C

•§

(G)=C

×C

(G)=C

=G/O

(G),G

=G/O

(G),G

=G/O

(G),

∼

l

ζ(G

)=C

,ζ(G

)=C

,ζ(G

)=1.

DOI:10.12677/pm.2021.1181651478nØêÆ

•»CK§°?‰

dÚn4•

Aut

Col

(G)=Projlim

1≤i≤2

(Inn(G

))=Inn(G

)×Inn(G

)

∼

5¿ζ(G)=C

,Inn(G)=G/ζ(G)

∼

§

Aut

Col

(G)=Inn(G).

½n2G=<a,b>§Ù ¥a

§…a

=1=b

,r6≡1(modp

)…r

≡1(modp

),p≡

1(mod5)§p´Œu5ƒê§KAut

Col

(G)=Inn(G)"

"ùG=<a>o=(C

)×C

§u´

(G)=C

×C

(G)=C

=G/O

(G),G

=G/O

(G),G

=G/O

(G),

∼

l

ζ(G

)=C

,ζ(G

)=C

,ζ(G

)=1.

dÚn4•

Aut

Col

(G)=Projlim

1≤i≤2

(Inn(G

))=Inn(G

)×Inn(G

)

∼

5¿ζ(G)=C

,Inn(G)=G/ζ(G)

∼

§

Aut

Col

(G)=Inn(G).

½n3G=<a,b,c>§Ù¥a

−1

§…a

=1=[a,c]=[b,c]§p´Œu5ƒ

ê§KAut

Col

(G)=Inn(G)"

"ùG=(<a>o)×<c>=

)×C

"u´

(G)=C

×C

(G)=C

=G/O

(G),G

=G/O

(G),G

=G/O

(G),

DOI:10.12677/pm.2021.1181651479nØêÆ

•»CK§°?‰

∼

l

ζ(G

)=C

,ζ(G

)=1,ζ(G

)=C

dÚn4•

Aut

Col

(G)=Projlim

1≤i≤2

(Inn(G

))=Inn(G

)×Inn(G

)

∼

5¿ζ(G)=C

,Inn(G)=G/ζ(G)

∼

,

Aut

Col

(G)=Inn(G).

Ä7‘8

I[g,‰ÆÄ7(11871292)"

ë•©z

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GroupsAreNilpotent.MathematischeZeitschrift,141,57-76.

https://doi.org/10.1007/BF01236984

[2]Hertweck,M.andKimmerle,W.(2002)ColemanAutomorphismsofFiniteGroups.Mathe-

matischeZeitschrift,242,203-215.https://doi.org/10.1007/s002090100318

[3]VanAntwerpen,A.(2018)ColemanAutomorphismsofFiniteGroupsandTheirMinimal

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https://doi.org/10.1016/j.jpaa.2017.12.013

‡),2017,52(10):4-6.

[5]Kimmerle,W.andRoggenkamp,K.W.(1993)ProjectiveLimitsofGroupRings.Journalof

PureandAppliedAlgebra,88,119-142.https://doi.org/10.1016/0022-4049(93)90017-N

[6]ÇöÀ,°?‰.2Â¡N+ColemangÓ+[J].ìÀŒÆÆ(nÆ‡),2020,55(12):

37-39.

[7]3¡ï,±•.Sylowp-f+•Ì‚+10p

š†+š†+[J].ÜHŒÆÆ(g,‰

Æ‡),2013,35(10):56-59.

[8]°?‰,½aû.ColemangÓ+Ý4•[J].êÆÆ,2020,63(4):281-288.

[9]M²„.k•+Ú[M].®:‰ÆÑ‡,1999.

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•»CK§°?‰

[10]Hai,J.K.andZhu,Y.X.(2018)OnColemanAutomorphismsofExtensionsofFinite

QuasinilpotentGroupsbySomeGroups.AlgebraColloquium,25,181-188.

https://doi.org/10.1142/S1005386718000123

DOI:10.12677/pm.2021.1181651481nØêÆ