除了無限系的單群之外，還有26個散在單群，早在1861年及1873年馬蒂厄已經知道其中的5個，1966-1975年間又陸續發現了21個。
到了1980年初，所有這26個零散單群都已經具體造出來。
gap> B:=SimpleGroup("B");Order(B);
B
4154781481226426191177580544000000
gap> ON:=SimpleGroup("ON");Order(ON);
ON
460815505920
gap> Suz:=SimpleGroup("Suz");Order(Suz);
Suz
448345497600
gap> Fi24:=SimpleGroup("Fi24");Order(Fi24);
Fi24'
1255205709190661721292800
gap> Co3:=SimpleGroup("Co3");Order(Co3);
Co_3
495766656000
gap> J4:=SimpleGroup("J(4)");Order(J4);
J4
86775571046077562880
gap> M11:=SimpleGroup("M(11)");Order(M11);StructureDescription(M11);cl:=ConjugacyClasses(M11);;len:=Size(cl);for i in [1..len] do Print(Size(cl[i]),","); od;Print("\n");for i in [1..len] do Print(Order(Centralizer(cl[i])),","); od;
M11
7920
"M11"
10
1,165,440,1320,1584,720,720,990,990,990,
7920,48,18,6,5,11,11,8,8,8,
gap> n:=7920;;for i in [1..n] do if n mod i=0 then Print(i,",");fi;od;
1,2,3,4,5,6,8,9,10,11,12,15,16,18,20,22,24,30,33,36,40,44,45,48,55,60,66,72,80,88,90,99,110,120,132,144,165,176,180,198,220,240,264,330,360,396,440,495,528,660,720,792,880,990,1320,1584,1980,2640,3960,7920,
gap> G:=M11;L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,8,9,10,11,12,15,16,18,20,22,24,30,33,36,40,44,45,48,55,60,66,72,80,88,90,99,110,120,132,144,165,176,180,198,220,240,264,330,360,396,440,495,528,660,720,792,880,990,1320,1584,1980,2640,3960,7920];;for i in M do Print(Size(Positions(L,i)),","); od;
M11
1,165,440,990,1584,1320,1980,0,0,1440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
gap> sl2M11:=SylowSubgroup(M11,2);IdGroup(sl2M11);
Group([ (2,10)(3,11)(4,8)(6,9), (3,4)(5,7)(6,9)(8,11), (2,3,6,4,10,11,9,8)(5,7) ])
[ 16, 8 ]
gap> sl3M11:=SylowSubgroup(M11,3);IdGroup(sl3M11);
Group([ (3,7,6)(4,9,5)(8,11,10), (3,4,10)(5,11,6)(7,9,8) ])
[ 9, 2 ]
gap> sl5M11:=SylowSubgroup(M11,5);IdGroup(sl5M11);
Group([ (2,3,11,10,5)(4,7,8,9,6) ])
[ 5, 1 ]
gap> sl11M11:=SylowSubgroup(M11,11);IdGroup(sl11M11);
Group([ (1,6,2,10,3,4,8,7,11,9,5) ])
[ 11, 1 ]
GAP4中調用NumberSmallGroups(2160);NumberSmallGroups(20160);NumberSmallGroups(10080);NumberSmallGroups(6720);NumberSmallGroups(5040);NumberSmallGroups(4032);NumberSmallGroups(3360);NumberSmallGroups(2880);NumberSmallGroups(2520);NumberSmallGroups(2240);NumberSmallGroups(2016);IdGroup(AlternatingGroup(8));IdGroup(GL(4,2));IdGroup(PSL(3,4));等指令會出錯。
調用下列指令就正常：
gap> 20160/8;
2520
gap> SmallSimpleGroup(2520,1);
A7
gap> SmallSimpleGroup(2520,2);
fail
gap> NumberSmallGroups(1680);
1005
gap> Order(AlternatingGroup(8));
20160
gap> Order(GL(4,2));
20160
gap> Order(PSL(3,4));
20160
gap> SmallSimpleGroup(20160,1);
A8
gap> SmallSimpleGroup(20160,2);
PSL(3,4)
第1種20160階單群
gap> 1+315+1232+3780+1344+5040+5760+2688;
20160
gap> n:=20160;;for i in [1..n] do if n mod i=0 then Print(i,",");fi;od;
1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,28,30,32,35,36,40,42,45,48,56,60,63,64,70,72,80,84,90,96,105,112,120,126,140,144,160,168,180,192,210,224,240,252,280,288,315,320,336,360,420,448,480,504,560,576,630,672,720,840,960,1008,1120,1260,1344,1440,1680,2016,2240,2520,2880,3360,4032,5040,6720,10080,20160,
gap> G:=AlternatingGroup(8);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,28,30,32,35,36,40,42,45,48,56,60,63,64,70,72,80,84,90,96,105,112,120,126,140,144,160,168,180,192,210,224,240,252,280,288,315,320,336,360,420,448,480,504,560,576,630,672,720,840,960,1008,1120,1260,1344,1440,1680,2016,2240,2520,2880,3360,4032,5040,6720,10080,20160];;for i in M do Print(Size(Positions(L,i)),","); od;
Alt( [ 1 .. 8 ] )
1,315,1232,3780,1344,5040,5760,0,0,0,0,0,2688,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
或
gap> G:=GL(4,2);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,28,30,32,35,36,40,42,45,48,56,60,63,64,70,72,80,84,90,96,105,112,120,126,140,144,160,168,180,192,210,224,240,252,280,288,315,320,336,360,420,448,480,504,560,576,630,672,720,840,960,1008,1120,1260,1344,1440,1680,2016,2240,2520,2880,3360,4032,5040,6720,10080,20160];;for i in M do Print(Size(Positions(L,i)),","); od;
SL(4,2)
1,315,1232,3780,1344,5040,5760,0,0,0,0,0,2688,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
第2種20160階單群
gap> 1+315+2240+3780+8064+5760;
20160
gap> G:=PSL(3,4);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,28,30,32,35,36,40,42,45,48,56,60,63,64,70,72,80,84,90,96,105,112,120,126,140,144,160,168,180,192,210,224,240,252,280,288,315,320,336,360,420,448,480,504,560,576,630,672,720,840,960,1008,1120,1260,1344,1440,1680,2016,2240,2520,2880,3360,4032,5040,6720,10080,20160];;for i in M do Print(Size(Positions(L,i)),","); od;
Group([ (3,4,5)(7,9,8)(10,14,18)(11,17,20)(12,15,21)(13,16,19), (1,2,6,7,11,3,10)(4,14,8,15,16,20,13)
(5,18,9,19,21,17,12) ])
1,315,2240,3780,8064,0,5760,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
第3種20160階群
gap> Phi(20160);
4608
gap> G:=CyclicGroup(20160);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,28,30,32,35,36,40,42,45,48,56,60,63,64,70,72,80,84,90,96,105,112,120,126,140,144,160,168,180,192,210,224,240,252,280,288,315,320,336,360,420,448,480,504,560,576,630,672,720,840,960,1008,1120,1260,1344,1440,1680,2016,2240,2520,2880,3360,4032,5040,6720,10080,20160];;for i in M do Print(Size(Positions(L,i)),","); od;
1,1,2,2,4,2,6,4,6,4,4,6,8,8,6,8,12,8,12,8,16,24,12,16,12,24,16,24,16,36,32,24,24,32,24,24,32,48,48,32,36,48,48,64,48,48,64,48,96,64,72,96,96,144,128,96,96,96,192,128,144,192,192,144,192,192,192,256,288,384,288,384,384,384,576,768,576,768,768,1152,1152,1536,2304,4608,
第4種20160階群
gap> G:=DihedralGroup(20160);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,28,30,32,35,36,40,42,45,48,56,60,63,64,70,72,80,84,90,96,105,112,120,126,140,144,160,168,180,192,210,224,240,252,280,288,315,320,336,360,420,448,480,504,560,576,630,672,720,840,960,1008,1120,1260,1344,1440,1680,2016,2240,2520,2880,3360,4032,5040,6720,10080,20160];;for i in M do Print(Size(Positions(L,i)),","); od;
1,10081,2,2,4,2,6,4,6,4,4,6,8,8,6,8,12,8,12,8,16,24,12,16,12,24,16,24,16,36,0,24,24,32,24,24,32,48,48,32,36,48,48,64,48,48,0,48,96,64,72,96,96,144,0,96,96,96,0,128,144,192,0,144,192,192,192,0,288,384,288,0,384,384,576,0,576,0,768,0,1152,0,2304,0,
第5種20160階群
gap> G:=QuaternionGroup(20160);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,28,30,32,35,36,40,42,45,48,56,60,63,64,70,72,80,84,90,96,105,112,120,126,140,144,160,168,180,192,210,224,240,252,280,288,315,320,336,360,420,448,480,504,560,576,630,672,720,840,960,1008,1120,1260,1344,1440,1680,2016,2240,2520,2880,3360,4032,5040,6720,10080,20160];;for i in M do Print(Size(Positions(L,i)),","); od;
1,1,2,10082,4,2,6,4,6,4,4,6,8,8,6,8,12,8,12,8,16,24,12,16,12,24,16,24,16,36,0,24,24,32,24,24,32,48,48,32,36,48,48,64,48,48,0,48,96,64,72,96,96,144,0,96,96,96,0,128,144,192,0,144,192,192,192,0,288,384,288,0,384,384,576,0,576,0,768,0,1152,0,2304,0,
第6種20160階群
gap> 20160/6;
3360
gap> G:=U6_3360;L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,28,30,32,35,36,40,42,45,48,56,60,63,64,70,72,80,84,90,96,105,112,120,126,140,144,160,168,180,192,210,224,240,252,280,288,315,320,336,360,420,448,480,504,560,576,630,672,720,840,960,1008,1120,1260,1344,1440,1680,2016,2240,2520,2880,3360,4032,5040,6720,10080,20160];;for i in M do Print(Size(Positions(L,i)),","); od;
1,1,8,2,4,8,6,4,0,4,16,6,32,8,0,8,48,32,12,32,16,24,0,16,48,0,64,24,64,0,96,24,0,32,96,0,128,192,48,128,0,48,0,64,192,0,192,192,96,256,0,96,0,0,384,384,0,384,576,512,0,192,0,0,768,0,768,768,0,384,0,1152,0,1536,0,2304,0,0,3072,0,0,4608,0,0,
第1種2520階群
gap> n:=2520;;for i in [1..n] do if n mod i=0 then Print(i,",");fi;od;
1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520,
gap> G:=AlternatingGroup(7);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520];;for i in M do Print(Size(Positions(L,i)),","); od;
Alt( [ 1 .. 7 ] )
1,105,350,630,504,210,720,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
第2種2520階群
gap> Phi(2520);
576
gap> G:=CyclicGroup(2520);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520];;for i in M do Print(Size(Positions(L,i)),","); od;
1,1,2,2,4,2,6,4,6,4,4,6,8,6,8,12,8,12,8,24,12,16,12,24,24,16,36,24,24,24,24,48,32,36,48,48,48,48,72,96,144,96,96,144,144,192,288,576,
第3種2520階群
gap> G:=DihedralGroup(2520);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520];;for i in M do Print(Size(Positions(L,i)),","); od;
1,1261,2,2,4,2,6,0,6,4,4,6,8,6,8,12,0,12,8,24,12,0,12,24,0,16,36,24,0,24,24,48,0,36,48,0,48,48,72,0,144,0,96,0,144,0,288,0,
第4種2520階群
gap> G:=QuaternionGroup(2520);L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520];;for i in M do Print(Size(Positions(L,i)),","); od;
1,1,2,1262,4,2,6,0,6,4,4,6,8,6,8,12,0,12,8,24,12,0,12,24,0,16,36,24,0,24,24,48,0,36,48,0,48,48,72,0,144,0,96,0,144,0,288,0,
第5種2520階群
gap> 2520/6;
420
gap> F:=FreeGroup(2);;U6_420:=F/[F.1^840,F.2^3,F.1^(-1)*F.2*F.1*F.2];;Order(U6_420);
2520
gap> G:=U6_420;L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520];;for i in M do Print(Size(Positions(L,i)),","); od;
1,1,8,2,4,8,6,12,0,4,16,6,32,0,8,48,24,12,32,24,0,48,48,0,72,64,0,24,0,96,0,192,96,0,48,144,0,192,0,288,0,0,384,0,0,576,0,0,
第6種2520階群
gap> 2520/8;
315
gap> F:=FreeGroup(2);;V8_315:=F/[F.1^630,F.2^4,F.2*F.1*(F.1^(-1)*F.2^(-1))^(-1),F.2^(-1)*F.1*(F.1^(-1)*F.2)^(-1)];;Order(V8_315);
2520
gap> G:=V8_315;L:=List(Elements(G),Order);;M:=[1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520];;for i in M do Print(Size(Positions(L,i)),","); od;
1,633,2,630,4,6,6,0,6,12,0,18,8,18,0,12,0,0,24,24,0,0,36,24,0,0,36,72,0,0,72,48,0,108,0,0,0,144,0,0,144,0,0,0,432,0,0,0,
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