50种72阶群

在陈松良等人的《关于72阶群的同构分类》一文中证明了G72共有50=10+4+32+4种不同构的类型:若Sylow子群都正规,则G72有10种;若Sylow 2-子群正规而Sylow 3-子群不正规,则G72有4种;若Sylow 3-子群正规而Sylow 2-子群不正规,则G72有32种;若Sylow子群都不正规,则G72有4种。
20151101猜想：有理数域上的分圆扩张的伽罗瓦群不可能是C24 x C3。或者说对任意n，(Z/nZ)^*≠C24 x C3。
互不同构的72阶交换群共有6个：
gap> G:=DirectProduct(CyclicGroup(3),CyclicGroup(24));;IdGroup(G);StructureDescription(G);
[ 72, 14 ]
"C24 x C3"
gap> n:=72;;for i in [n..600] do Ui:=Units(Integers mod i);;gid:=IdGroup(Ui);if n=gid[1] then Print(i,":",gid,"\n");fi;od;
73:[ 72, 2 ]
gap> G:=CyclicGroup(72);;IdGroup(G);StructureDescription(G);
[ 72, 2 ]
"C72"
91:[ 72, 36 ]
gap> G:=DirectProduct(CyclicGroup(6),CyclicGroup(12));;IdGroup(G);StructureDescription(G);
[ 72, 36 ]
"C12 x C6"
95:[ 72, 9 ]
gap> G:=DirectProduct(CyclicGroup(2),CyclicGroup(36));;IdGroup(G);StructureDescription(G);
[ 72, 9 ]
"C36 x C2"
111:[ 72, 9 ]
117:[ 72, 36 ]
135:[ 72, 9 ]
146:[ 72, 2 ]
148:[ 72, 9 ]
152:[ 72, 18 ]
gap> G:=DirectProduct(CyclicGroup(2),CyclicGroup(2),CyclicGroup(18));;IdGroup(G);StructureDescription(G);
[ 72, 18 ]
"C18 x C2 x C2"
182:[ 72, 36 ]
190:[ 72, 9 ]
216:[ 72, 18 ]
222:[ 72, 9 ]
228:[ 72, 18 ]
234:[ 72, 36 ]
252:[ 72, 50 ]
gap> G:=DirectProduct(CyclicGroup(2),CyclicGroup(6),CyclicGroup(6));;IdGroup(G);StructureDescription(G);
[ 72, 50 ]
"C6 x C6 x C2"
270:[ 72, 9 ]
gap> NumberSmallGroups(72);
50
gap>  for n in [1..50] do G:=SmallGroup(72,n);idn:=IdGroup(G);Print(idn);Print(":");L:=List(Elements(G),Order);;M:=[1,2,3,4,6,8,9,12,18,24,36,72];;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;
[ 72, 1 ]:1,1,2,2,2,36,6,4,6,0,12,0,
[ 72, 2 ]:1,1,2,2,2,4,6,4,6,8,12,24,
[ 72, 3 ]:1,1,2,6,2,0,24,12,24,0,0,0,
[ 72, 4 ]:1,1,2,38,2,0,6,4,6,0,12,0,
[ 72, 5 ]:1,19,2,20,2,0,6,4,6,0,12,0,
[ 72, 6 ]:1,37,2,2,2,0,6,4,6,0,12,0,
[ 72, 7 ]:1,3,2,36,6,0,6,0,18,0,0,0,
[ 72, 8 ]:1,21,2,18,6,0,6,0,18,0,0,0,
[ 72, 9 ]:1,3,2,4,6,0,6,8,18,0,24,0,
[ 72, 10 ]:1,5,2,2,10,0,6,4,30,0,12,0,
[ 72, 11 ]:1,1,2,6,2,0,6,12,6,0,36,0,
[ 72, 12 ]:1,1,8,2,8,12,0,16,0,24,0,0,
[ 72, 13 ]:1,1,8,2,8,36,0,16,0,0,0,0,
[ 72, 14 ]:1,1,8,2,8,4,0,16,0,32,0,0,
[ 72, 15 ]:1,21,2,18,6,0,24,0,0,0,0,0,
[ 72, 16 ]:1,7,2,0,14,0,24,0,24,0,0,0,
[ 72, 17 ]:1,39,2,0,6,0,6,0,18,0,0,0,
[ 72, 18 ]:1,7,2,0,14,0,6,0,42,0,0,0,
[ 72, 19 ]:1,1,8,18,8,36,0,0,0,0,0,0,
[ 72, 20 ]:1,7,8,24,20,0,0,12,0,0,0,0,
[ 72, 21 ]:1,19,8,12,8,0,0,24,0,0,0,0,
[ 72, 22 ]:1,13,8,18,32,0,0,0,0,0,0,0,
[ 72, 23 ]:1,25,8,6,20,0,0,12,0,0,0,0,
[ 72, 24 ]:1,1,8,30,8,0,0,24,0,0,0,0,
[ 72, 25 ]:1,1,26,6,26,0,0,12,0,0,0,0,
[ 72, 26 ]:1,1,8,14,8,0,0,40,0,0,0,0,
[ 72, 27 ]:1,7,8,8,20,0,0,28,0,0,0,0,
[ 72, 28 ]:1,13,8,2,32,0,0,16,0,0,0,0,
[ 72, 29 ]:1,3,8,12,24,0,0,24,0,0,0,0,
[ 72, 30 ]:1,9,8,6,36,0,0,12,0,0,0,0,
[ 72, 31 ]:1,1,8,38,8,0,0,16,0,0,0,0,
[ 72, 32 ]:1,19,8,20,8,0,0,16,0,0,0,0,
[ 72, 33 ]:1,37,8,2,8,0,0,16,0,0,0,0,
[ 72, 34 ]:1,3,8,36,24,0,0,0,0,0,0,0,
[ 72, 35 ]:1,21,8,18,24,0,0,0,0,0,0,0,
[ 72, 36 ]:1,3,8,4,24,0,0,32,0,0,0,0,
[ 72, 37 ]:1,5,8,2,40,0,0,16,0,0,0,0,
[ 72, 38 ]:1,1,8,6,8,0,0,48,0,0,0,0,
[ 72, 39 ]:1,9,8,18,0,36,0,0,0,0,0,0,
[ 72, 40 ]:1,21,8,18,24,0,0,0,0,0,0,0,
[ 72, 41 ]:1,9,8,54,0,0,0,0,0,0,0,0,
[ 72, 42 ]:1,9,26,6,18,0,0,12,0,0,0,0,
[ 72, 43 ]:1,21,26,18,6,0,0,0,0,0,0,0,
[ 72, 44 ]:1,15,26,0,30,0,0,0,0,0,0,0,
[ 72, 45 ]:1,19,8,36,8,0,0,0,0,0,0,0,
[ 72, 46 ]:1,31,8,0,32,0,0,0,0,0,0,0,
[ 72, 47 ]:1,7,26,0,38,0,0,0,0,0,0,0,
[ 72, 48 ]:1,15,8,0,48,0,0,0,0,0,0,0,
[ 72, 49 ]:1,39,8,0,24,0,0,0,0,0,0,0,
[ 72, 50 ]:1,7,8,0,56,0,0,0,0,0,0,0,
gap> for n in [1..50] do G:=SmallGroup(72,n);idn:=IdGroup(G);Print(idn);Print(":");L:=List(Elements(G),Order);;M:=[1,2,3,4,6,8,9,12,18,24,36,72];;for i in M do Print(Size(Positions(L,i)),","); od;arr:=[];;idn:=IdGroup(G);cl:=ConjugacyClasses(G);;Append(arr,"共轭类数：");;Append(arr,String(Size(cl)));Append(arr,"中心：");;Append(arr,String(IdGroup(Center(G))));;Append(arr,"换位子群：");;Append(arr,String(IdGroup(DerivedSubgroup(G))));;Append(arr,"自同构群：");;Append(arr,String(Order(AutomorphismGroup(G))));;cl:=NormalSubgroups(G);;Append(arr,"正规子群个数：");;len:=Size(cl);;Append(arr,String(len));;Print(arr);Print("\n");od;
[ 72, 1 ]:1,1,2,2,2,36,6,4,6,0,12,
0,共轭类数：24中心：[ 4, 1 ]换位子群：[ 9, 1 ]自同构群：216正规子群个数：10
[ 72, 2 ]:1,1,2,2,2,4,6,4,6,8,12,
24,共轭类数：72中心：[ 72, 2 ]换位子群：[ 1, 1 ]自同构群：24正规子群个数：12
[ 72, 3 ]:1,1,2,6,2,0,24,12,24,0,0,
0,共轭类数：21中心：[ 6, 2 ]换位子群：[ 8, 4 ]自同构群：72正规子群个数：7
[ 72, 4 ]:1,1,2,38,2,0,6,4,6,0,12,
0,共轭类数：21中心：[ 2, 1 ]换位子群：[ 18, 2 ]自同构群：432正规子群个数：12
[ 72, 5 ]:1,19,2,20,2,0,6,4,6,0,12,
0,共轭类数：24中心：[ 4, 1 ]换位子群：[ 9, 1 ]自同构群：216正规子群个数：14
[ 72, 6 ]:1,37,2,2,2,0,6,4,6,0,12,
0,共轭类数：21中心：[ 2, 1 ]换位子群：[ 18, 2 ]自同构群：432正规子群个数：12
[ 72, 7 ]:1,3,2,36,6,0,6,0,18,0,0,
0,共轭类数：24中心：[ 4, 2 ]换位子群：[ 9, 1 ]自同构群：432正规子群个数：18
[ 72, 8 ]:1,21,2,18,6,0,6,0,18,0,0,
0,共轭类数：21中心：[ 2, 1 ]换位子群：[ 18, 2 ]自同构群：216正规子群个数：12
[ 72, 9 ]:1,3,2,4,6,0,6,8,18,0,24,
0,共轭类数：72中心：[ 72, 9 ]换位子群：[ 1, 1 ]自同构群：48正规子群个数：24
[ 72, 10 ]:1,5,2,2,10,0,6,4,30,0,12,
0,共轭类数：45中心：[ 18, 2 ]换位子群：[ 2, 1 ]自同构群：48正规子群个数：18
[ 72, 11 ]:1,1,2,6,2,0,6,12,6,0,36,
0,共轭类数：45中心：[ 18, 2 ]换位子群：[ 2, 1 ]自同构群：144正规子群个数：18
[ 72, 12 ]:1,1,8,2,8,12,0,16,0,24,0,
0,共轭类数：36中心：[ 12, 2 ]换位子群：[ 3, 1 ]自同构群：48正规子群个数：14
[ 72, 13 ]:1,1,8,2,8,36,0,16,0,0,0,
0,共轭类数：24中心：[ 4, 1 ]换位子群：[ 9, 2 ]自同构群：1728正规子群个数：19
[ 72, 14 ]:1,1,8,2,8,4,0,16,0,32,0,
0,共轭类数：72中心：[ 72, 14 ]换位子群：[ 1, 1 ]自同构群：192正规子群个数：24
[ 72, 15 ]:1,21,2,18,6,0,24,0,0,0,0,
0,共轭类数：9中心：[ 1, 1 ]换位子群：[ 36, 3 ]自同构群：216正规子群个数：6
[ 72, 16 ]:1,7,2,0,14,0,24,0,24,0,0,
0,共轭类数：24中心：[ 6, 2 ]换位子群：[ 4, 2 ]自同构群：72正规子群个数：10
[ 72, 17 ]:1,39,2,0,6,0,6,0,18,0,0,
0,共轭类数：24中心：[ 4, 2 ]换位子群：[ 9, 1 ]自同构群：1296正规子群个数：26
[ 72, 18 ]:1,7,2,0,14,0,6,0,42,0,0,
0,共轭类数：72中心：[ 72, 18 ]换位子群：[ 1, 1 ]自同构群：1008正规子群个数：48
[ 72, 19 ]:1,1,8,18,8,36,0,0,0,0,0,
0,共轭类数：12中心：[ 2, 1 ]换位子群：[ 9, 2 ]自同构群：288正规子群个数：6
[ 72, 20 ]:1,7,8,24,20,0,0,12,0,0,0,
0,共轭类数：18中心：[ 2, 1 ]换位子群：[ 9, 2 ]自同构群：144正规子群个数：18
[ 72, 21 ]:1,19,8,12,8,0,0,24,0,0,0,
0,共轭类数：18中心：[ 2, 1 ]换位子群：[ 9, 2 ]自同构群：288正规子群个数：16
[ 72, 22 ]:1,13,8,18,32,0,0,0,0,0,0,
0,共轭类数：15中心：[ 2, 1 ]换位子群：[ 18, 5 ]自同构群：288正规子群个数：14
[ 72, 23 ]:1,25,8,6,20,0,0,12,0,0,0,
0,共轭类数：15中心：[ 2, 1 ]换位子群：[ 18, 5 ]自同构群：144正规子群个数：14
[ 72, 24 ]:1,1,8,30,8,0,0,24,0,0,0,
0,共轭类数：15中心：[ 2, 1 ]换位子群：[ 18, 5 ]自同构群：288正规子群个数：14
[ 72, 25 ]:1,1,26,6,26,0,0,12,0,0,0,
0,共轭类数：21中心：[ 6, 2 ]换位子群：[ 8, 4 ]自同构群：144正规子群个数：10
[ 72, 26 ]:1,1,8,14,8,0,0,40,0,0,0,
0,共轭类数：27中心：[ 6, 2 ]换位子群：[ 6, 2 ]自同构群：96正规子群个数：18
[ 72, 27 ]:1,7,8,8,20,0,0,28,0,0,0,
0,共轭类数：36中心：[ 12, 2 ]换位子群：[ 3, 1 ]自同构群：48正规子群个数：22
[ 72, 28 ]:1,13,8,2,32,0,0,16,0,0,0,
0,共轭类数：27中心：[ 6, 2 ]换位子群：[ 6, 2 ]自同构群：96正规子群个数：18
[ 72, 29 ]:1,3,8,12,24,0,0,24,0,0,0,
0,共轭类数：36中心：[ 12, 5 ]换位子群：[ 3, 1 ]自同构群：96正规子群个数：26
[ 72, 30 ]:1,9,8,6,36,0,0,12,0,0,0,
0,共轭类数：27中心：[ 6, 2 ]换位子群：[ 6, 2 ]自同构群：48正规子群个数：18
[ 72, 31 ]:1,1,8,38,8,0,0,16,0,0,0,
0,共轭类数：21中心：[ 2, 1 ]换位子群：[ 18, 5 ]自同构群：3456正规子群个数：21
[ 72, 32 ]:1,19,8,20,8,0,0,16,0,0,0,
0,共轭类数：24中心：[ 4, 1 ]换位子群：[ 9, 2 ]自同构群：1728正规子群个数：23
[ 72, 33 ]:1,37,8,2,8,0,0,16,0,0,0,
0,共轭类数：21中心：[ 2, 1 ]换位子群：[ 18, 5 ]自同构群：3456正规子群个数：21
[ 72, 34 ]:1,3,8,36,24,0,0,0,0,0,0,
0,共轭类数：24中心：[ 4, 2 ]换位子群：[ 9, 2 ]自同构群：3456正规子群个数：33
[ 72, 35 ]:1,21,8,18,24,0,0,0,0,0,0,
0,共轭类数：21中心：[ 2, 1 ]换位子群：[ 18, 5 ]自同构群：1728正规子群个数：21
[ 72, 36 ]:1,3,8,4,24,0,0,32,0,0,0,
0,共轭类数：72中心：[ 72, 36 ]换位子群：[ 1, 1 ]自同构群：384正规子群个数：48
[ 72, 37 ]:1,5,8,2,40,0,0,16,0,0,0,
0,共轭类数：45中心：[ 18, 5 ]换位子群：[ 2, 1 ]自同构群：384正规子群个数：36
[ 72, 38 ]:1,1,8,6,8,0,0,48,0,0,0,
0,共轭类数：45中心：[ 18, 5 ]换位子群：[ 2, 1 ]自同构群：1152正规子群个数：36
[ 72, 39 ]:1,9,8,18,0,36,0,0,0,0,0,
0,共轭类数：9中心：[ 1, 1 ]换位子群：[ 9, 2 ]自同构群：144正规子群个数：5
[ 72, 40 ]:1,21,8,18,24,0,0,0,0,0,0,
0,共轭类数：9中心：[ 1, 1 ]换位子群：[ 18, 4 ]自同构群：144正规子群个数：7
[ 72, 41 ]:1,9,8,54,0,0,0,0,0,0,0,
0,共轭类数：6中心：[ 1, 1 ]换位子群：[ 18, 4 ]自同构群：432正规子群个数：7
[ 72, 42 ]:1,9,26,6,18,0,0,12,0,0,0,
0,共轭类数：15中心：[ 3, 1 ]换位子群：[ 12, 3 ]自同构群：48正规子群个数：8
[ 72, 43 ]:1,21,26,18,6,0,0,0,0,0,0,
0,共轭类数：9中心：[ 1, 1 ]换位子群：[ 36, 11 ]自同构群：432正规子群个数：9
[ 72, 44 ]:1,15,26,0,30,0,0,0,0,0,0,
0,共轭类数：12中心：[ 1, 1 ]换位子群：[ 12, 5 ]自同构群：144正规子群个数：9
[ 72, 45 ]:1,19,8,36,8,0,0,0,0,0,0,
0,共轭类数：12中心：[ 2, 1 ]换位子群：[ 9, 2 ]自同构群：288正规子群个数：10
[ 72, 46 ]:1,31,8,0,32,0,0,0,0,0,0,
0,共轭类数：18中心：[ 2, 1 ]换位子群：[ 9, 2 ]自同构群：288正规子群个数：28
[ 72, 47 ]:1,7,26,0,38,0,0,0,0,0,0,
0,共轭类数：24中心：[ 6, 2 ]换位子群：[ 4, 2 ]自同构群：144正规子群个数：16
[ 72, 48 ]:1,15,8,0,48,0,0,0,0,0,0,
0,共轭类数：36中心：[ 12, 5 ]换位子群：[ 3, 1 ]自同构群：288正规子群个数：42
[ 72, 49 ]:1,39,8,0,24,0,0,0,0,0,0,
0,共轭类数：24中心：[ 4, 2 ]换位子群：[ 9, 2 ]自同构群：10368正规子群个数：41
[ 72, 50 ]:1,7,8,0,56,0,0,0,0,0,0,
0,共轭类数：72中心：[ 72, 50 ]换位子群：[ 1, 1 ]自同构群：8064正规子群个数：96