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PureMathematicsnØêÆ,2021,11(12),1993-2002
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1112222
Ýa•üü•ŒúÏf–õkü‡ƒÏf
k•ü+
ÜÜ܈ˆˆ•••
1
§§§444ÿÿÿddd
2
1
Êô…’ŒÆ“‰Æ§ôÜÊô
2
ôÜ“‰ŒÆêƆÚOÆ§ôÜH
ÂvFϵ2021c114F¶¹^Fϵ2021c126F¶uÙFϵ2021c1213F
Á‡
©y²A
5
´•˜÷v?¿ü‡ØÓÝa•• ŒúÏf–õkü‡(ؘ½ØÓ)ƒÏf
k•ü+"
'…c
k•ü+§Ýa§•ŒúÏf§ƒÏf
FiniteSimpleGroupsinWhichAnyTwo
DifferentConjugacyClassLengthsHave
atMostTwoPrimeDivisorsinCommon
YaofangZhang
1
,YanjunLiu
2
1
NormalSchool,JiujiangVocationalUniversity,JiujiangJiangxi
2
SchoolofMathematicsandStatistics,JiangxiNormalUniversity,NanchangJiangxi
Received:Nov.4
th
,2021;accepted:Dec.6
th
,2021;published:Dec.13
th
,2021
©ÙÚ^:܈•,4ÿd.Ýa•üü•ŒúÏf–õkü‡ƒÏfk•ü+[J].nØêÆ,2021,11(12):
1993-2002.DOI:10.12677/pm.2021.1112222
܈•§4ÿd
Abstract
ThispapershowsthatA
5
istheonlyfinitesimplegroupsuchthatthegreatestcom-
mondivisorofanypairofitsdifferentconjugacyclasslengthshasatmosttwo(not
necessarilydifferent)primedivisors.
Keywords
FiniteSimpleGroup,ConjugacyClass,GreatestCommonDivisor,PrimeDivisor
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
k•+Ø´˜‡š~¹ïÄ+•,E,„kNõ™)û¯K.¯¢þgl1965cm©,z
…AcÛd‰ÆêƤѬч+Ø¥™)ûúm¯K8,y®²ò C20‡,•„[1].Ï
~,k•+AInØ¥AIÝê†+Ýa•kXéõ™éÐn)éX.3˜½¿Âþ,
Ýa•Œ±wŠÚAIÝêéó.G•k•+,·‚¡8Ücd(G)={χ(1)|χ∈Irr(G)}•
GEØŒAIÝê8Ü.3k•+AInØïÄ¥,b½G?¿ü‡ØÓØŒAI
Ýêpƒ,¯¤±•,cd(G) ≤3 XJGŒ).éu˜„œ/,Šâk•+AIÝêãëÏ©|
ê½n,·‚kcd(G) ≤4 .
fzpƒ^‡,M.lewis ÇkïÄ÷vüƒêbk•+,ùpüƒêb•éuØÓÝ
êχ(1) ,ψ(1)∈cd(G) ,•ŒúÏfgcd(χ(1),ψ(1)) ‡o•1,‡o•,˜ƒê.3XØ©¥,¦
y²÷vüƒêbk•+G,k|cd(G)|≤9.÷Xù‡ ••,M.Lewis 3©z[2]¥Ú\
n−ƒêbù˜Vg,äN5`,¡G÷vn−ƒêb,eéuØÓÝêχ(1) ,ψ(1)∈cd(G)
,gcd(χ(1),ψ(1)) ؘ½ØÓƒÏfo‡ê–õ•n.3T©¥,ŠöJÑeãߎ:GŒ
)ž,•3½Â3šKêþꊼêf(n) ,¦|cd(G)|≤f(n).ù˜ßŽyÜ©y²
´é,¿…n=0,1 ž,f(0)=3,f(1)=9 ,éun=2,J.Hamblin †M.Lewisy²
f(2) ≤462515,•„[3].
2010c,4ÿd,yÆ ÚÜU²3©z[4]¥ïÄ÷vƒê•˜bšŒ)+,ùpƒê
•˜b•,éuØÓÝêχ(1) ,ψ(1)∈cd(G) ,ö•ŒúÏfgcd(χ(1),ψ(1)) •ƒê•˜.
2017c,ÚVÚMarkL.Lewis 3©z[5]¥ïÄ÷vƒê•˜Œ)+,(صXJG•÷
DOI:10.12677/pm.2021.11122221994nØêÆ
܈•§4ÿd
vƒê•˜Œ)+,KGFitting p–õ•12.XJ|G|•Ûê…G÷vƒê•˜b,KG
Fittingp–õ•6.
CÏ,CaminaIåïÄ÷v?¿ü‡ØÓÝa••ŒúÏfþ•ƒê•˜šŒ)+,
˜éÐ5Ÿ§•„[6].•d,©ïÄ÷v?¿ü‡ØÓÝa••ŒúÏf–õ
kü‡£Ø˜½ØÓ¤ƒÏfk•ü+,̇(ØXeµ
½n1.S•š†ü+.eS?¿ü‡ØÓÝa••ŒúÏf•õkü‡(Œ±ƒÓ)
ƒÏf,KS•A
1
(4)
∼
=
A
1
(5)
∼
=
A
5
.
2.ý•£
!k0Ø©¤I‡˜{ü(Ø.
Ún1.G=SL(2,F),Ù¥F•kq=p
n
‡ƒk••,p´Ûƒê.-v•Ì‚+
F
∗
= F−{0})¤,KG¥kƒ
1 =
10
01
!
,z=
−10
0−1
!
,c=
10
11
!
,d=
10
v1
!
,a=
v0
0v
−1
!
,
|a|=q−1 ,…G•¹˜‡ •q+ 1ƒb.é∀x∈G,(x) •G¥•¹xÝa,Gk
q+4 ‡Ýa,©O•µ
(1),(z) ,(c) ,(d),(zc) ,(zd) ,(a) ,(a
2
) ,···,(a
(q −3)/2
) ,(b) ,(b
2
) ,···,b
(q −1)/2
,
ÙÝa••L1
Table1.ConjugateclasslengthsofSL(2,F),podd
L1.SL(2,F)Ýa•§pÛê
x1zcdzczda
l
b
m
|x|11
1
2
(q
2
−1)
1
2
(q
2
−1)
1
2
(q
2
−1)
1
2
(q
2
−1)q(q+1)q(q−1)
Ù¥:1 ≤l≤(q−3)/2,1 ≤m≤(q−1)/2.
y²:•„([7],½n38.1).2
Ún2.G= SL(2,F) ,Ù¥F•kq= 2
n
‡ƒk••.-v•Ì‚+F
∗
= F−{0})
¤,KG¥kƒ:
1 =
10
01
!
,c=
10
11
!
,a=
v0
0v
−1
!
,
…G¥•¹˜‡•q+1 ƒb.é∀x∈G,(x) •G¥•¹xÝa,Gkq+1 ‡Ý
a,©O•:
(1) ,(c) ,(a) ,(a
2
) ,···,(a
(q −2)/2
) ,(b) ,(b
2
) ,···,(b
q /2
) ,
ÙÝa••„L2,
DOI:10.12677/pm.2021.11122221995nØêÆ
܈•§4ÿd
Table2.ConjugateclasslengthsofSL(2,F),peven
L2.SL(2,F)Ýa•§póê
x1ca
l
b
m
|x|1(q
2
−1)q(q+1)q(q−1)
Ù¥:1 ≤l≤(q−2)/2,1 ≤m≤q/2 .
y²:•„([7],½n38.2).2
Ún3.G= SL
2
(q) ,G/Z= PSL
2
(q) ,K
|G: C
G
(x)|= |G: C
G
(x)|,
=±xÚx•“Lƒü‡Ýa•ÝƒÓ.
y²:†Ž=.2
3.k•ü+Ýa•
Šâü+©a½n§k•š†ü+•±e+ƒ˜µlÑü+!†+!o.ü+±9Tits
ü+§•„[8].
3.1.†+
½n1.S=A
n
(n≥5) ,e+S?¿ü‡ØÓÝa••Œ úÏf•õkü‡(Œ± ƒÓ)
ƒÏf,KS= A
5
.
y²:n= 5ž,A
5
ØÓÝa•©O•1,12,15,20.§‚•ŒúÏf©O•µ
(12,15) = 3,(12,20) = 2
2
,(15,20) = 5.
¤±A
5
?¿ü‡ØÓÝa••ŒúÏf•õkü‡(Œ±ƒÓ)ƒÏf.
n= 6ž,A
6
ØÓÝa•©O•1,40,45,72,90.§‚•ŒúÏf©O•:
(40,45) = 5,(40,72) = 2
3
,(40,90) = 2·5,
(45,72) = 3
2
,(45,90) = 3
2
·5,(72,90) = 2·3
2
.
Ù¥,±(12)(34) •“LƒÝa••45,±(1234)(56) •“LƒÝa••90.¤±A
6
kü‡ØÓÝa•§Ù•ŒúÏfk3‡(Œ±ƒÓ)ƒÏf.
n= 7ž,A
7
ØÓÝa•©O•:
1,70,105,210,280,360,504,630.
DOI:10.12677/pm.2021.11122221996nØêÆ
܈•§4ÿd
70Ú210•ŒúÏf•70=2 ·5·7 .Ù¥,±(123) •“LƒÝ a••70,±(123)(45)(67)
•“LƒÝa••210,¤±A
7
kü‡ØÓÝa••ŒúÏfk3‡(Œ±ƒÓ)ƒÏf.
n= 8ž,A
8
¥±(123)(456) •“LƒÝa•´1120,±(123456)(78) •“Lƒ
Ýa•´3360,
(1120,3360) = 1120,1120 = 2
3
·4·5·7,
A
8
kü‡ØÓÝa•§Ù•ŒúÏfk3‡±þ(Œ±ƒÓ)ƒÏf.n=9ž,A
9
¥±
(12345)•“LƒÝa•´3024,±(12345)(67)(89) •“LƒÝa•´9072,
(3024,9072) = 3024,3024 =3
3
·4
2
·7,
¤±A
9
kü‡ØÓÝa•§Ù•ŒúÏfk3‡±þ(Œ±ƒÓ)ƒÏf.
aquþ˜Ùé†+©Û,éuA
n
,n= 2k(k≥3) ž,•ÄÝa“Lƒ:
(1,2,···,k−1)(k,k+1,···,2k−2)9(1,2,···,2k−2)(2k−1,2k).
§‚Ýa•©O•
3·4·5···(k−2)·k·(k+1)···(2k−3)·(2k−1)·2k
9
3·4·5···k·(k+1)···(2k−3)·(2k−1)·2k,
Œ±wÑö´cö(k−1),§‚•ŒúÏf•:
2k·(2k−1)·(2k−3)···k·(k−2)···4·3.
´•,k≥3 ž,A
2k
kü‡ØÓÝa•§Ù•ŒúÏfk2‡±þ(Œ±ƒÓ)ƒÏf.
éuA
n
,n= 2k+1(k≥3) ž,•ÄÝa“Lƒ
(1,2,···,2k−3) 9(1,2,···,2k−3)(2k−2,2k−1)(2k,2k+1) .
§‚Ýa•©O•
(2k+1)·2k·(2k−1)·(2k−2)·(2k−4)···5
9
(2k+1)·2k·(2k−1)·(2k−2)·(2k−4)···5·3,
Œ±wÑö´cö3,§‚•ŒúÏf•
(2k+1)·2k·(2k−1)·(2k−2)·(2k−4)···5.
´•,k≥3ž,A
2k+1
kü‡ØÓÝa•§Ù•ŒúÏfk3‡±þ(Œ±ƒÓ) ƒÏf.½n
y.2
DOI:10.12677/pm.2021.11122221997nØêÆ
܈•§4ÿd
3.2.lÑü+
½n2.S•Titsü+½ö26‡lÑü+ƒ˜,KS–kü‡ØÓÝa•,§‚•ŒúÏ
fkn‡9±þ(Œ±ƒÓ)ƒÏf.
y²:ŠâGAP§S[9]=Œy,•„L3.
Table3.Maximumcommonfactorfordifferentconjugateclasslengths
L3.•ŒÝa•úÏf
lÑü+Ýa••ŒúÏflÑü+Ýa••ŒúÏf
M
11
2·3
2
·5Fi
0
24
2
2
·3
7
·7
2
·23·29
M
12
3
2
·11HS5
2
·11·7
M
22
5·7·11McL5
2
·11
M
23
3·5·11·23He3·5·7
2
·17
M
24
3
2
·11·23Ru3
2
·5
2
·29
J
1
7·11·19Suz3
3
·5·11·13
J
2
3
2
·5·7O
0
N7
2
·19·11·31
J
3
3
2
·17·19HN3
4
·5
3
·19
J
4
11
2
·23·29·31·37·43Ly5
3
·31·37·67
Co
1
3
4
·5
2
·7·11·23Th3
3
·5
2
·7·19·31
Co
2
3
2
·5
2
·11·23B3
4
·5
3
·31·47
Co
3
3
3
·5
2
·23M3
7
·5
3
·7
4
·11·13
2
·29·41·59·71
Fi
22
3
3
·5·13
2
F
4
(2)
0
3
2
·5·13
Fi
23
2
8
·5·11·13·17·23
3.3.o.ü+
!ïÄo.ü+,Ü©Ú^ÎÒë„[7]½[10].
·K1.S•~ü+,KS–kü‡ØÓÝa•,§‚•ŒúÏfkn‡9±þ(Œ±
ƒÓ)ƒÏf.
y²:z˜‡~ü+±s
1
,s
2
•“LƒÝa•kúÏfq
3
,=y.2
·K2.S•A
n
(q) ,Ù¥q•ƒê•˜,n∈Z
+
,(q>3 XJn=1) ,•kS=A
1
(4) Ú
A
1
(5) ž,Ù?¿ü‡ØÓÝa••ŒúÏf•õkü‡(Œ±ƒÓ)ƒÏf.
y²:5¿±s
1
•“LƒÝa••:
q
1
2
n(n+1)
(n+1,q−1)
n
Y
i=1
(q
i
−1),
±s
2
•“LƒÝa••:
q
1
2
n(n+1)
(n+1,q−1)
(q
n+1
−1)
n−2
Y
i=1
(q
i+1
−1).
DOI:10.12677/pm.2021.11122221998nØêÆ
܈•§4ÿd
n≥2ž,ùü‡Ýa•7kúÏfq
3
,¤±A
n
(q)(n≥2)–kü‡ØÓÝa•§Ù•Œ
úÏfk–n‡(Œ±ƒÓ)ƒÏf.
en= 1.
1.q•Ûƒê•˜=q6= 2
f
ž,ŠâÚn3†1Œ•,•k:
(q+1)(
1
2
(q−1),q),2q,(q−1)(
1
2
(q+1),q)
þ•õkü‡(ŒƒÓ) ƒÏfž,A
1
(q) Ù?¿ü‡ØÓÝa••ŒúÏf•õkü‡
(Œ±ƒÓ) ƒÏf.e2q•õkü‡(ŒƒÓ) ƒÏf,q•U•ƒê˜g˜,dž•I•Ä
q−1,q+1 ,
q= 3 ž,q+1 = 4 = 2
2
,q−1 = 2,
¤±A
1
(3) Ù?¿ü‡ØÓÝa••ŒúÏf•õkü‡(Œ±ƒÓ) ƒÏf.´A
1
(3)
Œ),Ø7•Ä.
q= 5 ž,q+1 = 6 = 2·3,q−1 = 4 = 2
2
,
¤±A
1
(5)Ù?¿ü‡ØÓÝa••ŒúÏf•õkü‡(Œ±ƒÓ)ƒÏf.
q≥7 ž,q+1,q−1 ¥–k˜‡k–n‡(ŒƒÓ) ƒÏf,
¤±A
1
(q)(q6= 2
f
,p≥7)ØÎÜ.
2.q=2
f
ž,ŠâÚn2Œ•,A
1
(q) ØÓÝa••ŒúÏf©O•q,q−1,q+1 ,Äk
•Äq,eq•õkü‡ƒÏf(Œ±ƒÓ) ,Kq= 2 ½2
2
,A
1
(2) Œ),Ø7•Ä.q=2
2
ž,q−1 = 3,q+1 = 5,džq,q−1,q+1þ•õkü‡(Œ±ƒÓ) ƒÏf,¤±A
1
(4)Ù
?¿ü‡ØÓÝa••ŒúÏf•õkü‡(Œ±ƒÓ)ƒÏf.
nþ¤ã,•kS=A
1
(4)
∼
=
A
1
(5) ž,Ù?¿ü‡ØÓÝa••Œ úÏf•õkü‡
(Œ±ƒÓ)ƒÏf.·Ky.2
·K3.S•
2
A
n
(q
2
) ,q•ƒê•˜,n∈Z
+
,Ù¥n≥2( ±9q>2 XJn= 2) ,KS–
kü‡ØÓÝa•,§‚•ŒúÏfkn‡9±þ(Œ±ƒÓ)ƒÏf.
y²:n≥3 •Ûêž,±s
1
•“LƒÝa••:
1
(n+1,q+1)
q
1
2
n(n+1)
n
Y
i=1
(q
i
−(−1)
i
),
±s
2
•“LƒÝa••:
(q
n+1
−1)
(n+1,q+1)
q
1
2
n(n+1)
n−2
Y
i=1
(q
i+1
−(−1)
i+1
).
ùü‡Ýa•k úÏfq
3
,dž
2
A
n
(q
2
) kü‡ØÓÝa••ŒúÏfkn‡9±þ(Œ
±ƒÓ)ƒÏf.
n≥2 •óêž,±s
1
•“LƒÝa•Ý•:
1
(n+1,q+1)
q
1
2
n(n+1)
n
Y
i=1
(q
i
−(−1)
i
),
DOI:10.12677/pm.2021.11122221999nØêÆ
܈•§4ÿd
±s
2
•“LƒÝa•Ý•:
(q
n+1
+1)
(n+1,q+1)
q
1
2
n(n+1)
n−2
Y
i=1
(q
i+1
−(−1)
i+1
).
ùü‡Ýa•kúÏfq
3
,Ódž
2
A
n
(q
2
)kü‡ØÓÝa••ŒúÏfkn‡9±þ
(Œ±ƒÓ)ƒÏf.·Ky.2
·K4.S•B
n
(q) ½C
n
(q),n≥2( ±9q>2 XJn= 2) ,Ù¥q•ƒê•˜,KS–
kü‡ØÓÝa•,§‚•ŒúÏf–kn‡(Œ±ƒÓ)ƒÏf.
y²:n≥3 •Ûêž,±s
1
•“LƒÝa••:
q
n
2
(2,q−1)
1
q
n
+1
n
Y
i=1
(q
2i
−1),
±s
2
•“LƒÝa••:
q
n
2
(2,q−1)
1
q
n
−1
n
Y
i=1
(q
2i
−1).
ùü‡Ýa•kúÏfq
3
,džSkü‡ØÓÝa••ŒúÏfkn‡9±þ(Œ±ƒ
Ó)ƒÏf.
n≥2 •óêž,±s
1
•“LƒÝa•Ý•
q
n
2
(2,q−1)
1
q
n
+1
n
Y
i=1
(q
2i
−1),
±s
2
•“LƒÝa•Ý•
q
n
2
(2,q−1)
1
(q+1)(q
n−1
+1)
n
Y
i=1
(q
2i
−1).
ùü‡Ýa•kúÏfq
3
,ÓdžSkü‡ØÓÝa••ŒúÏfkn‡ 9 ±þ(Œ±
ƒÓ)ƒÏf.·Ky.2
·K5.S•D
n
(q) ,Ù¥q•ƒê•˜,n>3,n∈Z
+
,KS–kü‡ØÓÝa•,§
‚•ŒúÏf–kn‡(Œ±ƒÓ)ƒÏf.
y²:n≥5 •Ûêž,±s
1
•“LƒÝa••
q
n(n−1)
(q
n
−1)
(4,q
n
−1)
1
(q
n−1
+1)(q+1)
n−1
Y
i=1
(q
2i
−1),
±s
2
•“LƒÝa••
q
n(n−1)
(q
n
−1)
(4,q
n
−1)
1
(q
n
−1)
n−1
Y
i=1
(q
2i
−1).
DOI:10.12677/pm.2021.11122222000nØêÆ
܈•§4ÿd
ùü‡Ýa•kúÏfq
3
,džSkü‡ ØÓÝa•§Ù•ŒúÏfkn‡9±þ(Œ±ƒ
Ó)ƒÏf.
n≥4 •óêž,±s
1
•“LƒÝa•Ý•
q
n(n−1)
(q
n
−1)
(4,q
n
−1)
1
(q
n−1
+1)(q+1)
n−1
Y
i=1
(q
2i
−1),
±s
2
•“LƒÝa•Ý•
q
n(n−1)
(q
n
−1)
(4,q
n
−1)
1
(q
n−1
−1)(q−1)
n−1
Y
i=1
(q
2i
−1).
ùü‡Ýa•kúÏfq
3
,ÓdžSkü‡ØÓÝa••ŒúÏfkn‡ 9 ±þ(Œ±
ƒÓ)ƒÏf.·Ky.2
·K6.S•
2
D
n
(q
2
) ,Ù¥q•ƒê•˜,n>3,n∈Z
+
,KS–kü‡ØÓÝa•,
§‚•ŒúÏfkn‡9±þ(Œ±ƒÓ)ƒÏf.
y²:±s
1
•“LƒÝa•Ý•
1
(4,q
n
+1)
q
n(n−1)
(q
n
+1)
1
q
n
+1
n−1
Y
i=1
(q
2i
−1),
±s
2
•“LƒÝa••
1
(4,q
n
+1)
q
n(n−1)
(q
n
+1)
1
(q
n−1
+1)(q−1)
n−1
Y
i=1
(q
2i
−1).
ùü‡Ýa•kúÏfq
3
,džSkü‡ØÓÝa••ŒúÏfkn‡9±þ(Œ±ƒ
Ó)ƒÏf.2
½n3.eS•o.ü+,eSØÓÝa••ŒúÏf• õk ü‡(Œ±ƒÓ)ƒÏf,K•
A
1
(4) ½A
1
(5) .
y²:Šâ·K1-6=Œy.2
½n1y²:Šâ½n1-3=Œy.
—
©ôÜŽ“c‰ÆÄ7))-:‘820192ACB21008]Ï,Ada!
ë•©z
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Notebook.arXiv:1401.0300[math.GR]
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܈•§4ÿd
[2]Lewis,M.(2005)The NumberofIrreducible CharacterDegreesof SolvableGroupsSatisfying
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Sizes.InternationalJournalofGroupTheory,6,13-19.
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MarcelDekker,NewYork.
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