有限羣秩、最小置換表示次數的計算

定理1：S_n(n>2)都是二元生成羣，即rank(S_n)=2，其生成元可取(1,2)及(2,3,…,n)。
定理2：S_n(n>3)的一元生成子羣均不是極大子羣。
GAP命令爲：
gap> n:=8;;len:=NumberSmallGroups(n);;for i in [1..len] do Print(Rank(SmallGroup(n,i)),","); od;
1,2,2,2,3
gap> n:=8;;len:=NumberSmallGroups(n);;for i in [1..len] do Print(RankPGroup(SmallGroup(n,i)),","); od;
1,2,2,2,3,
gap> n:=16;;len:=NumberSmallGroups(n);;for i in [1..len] do Print(Rank(SmallGroup(n,i)),","); od;
1,2,2,2,2,2,2,2,2,3,3,3,3,4,
gap> n:=8;;len:=NumberSmallGroups(n);;for i in [1..len] do Print(Size(SmallGeneratingSet(SmallGroup(n,i))),","); od;
1,2,2,2,3,
gap> n:=16;;len:=NumberSmallGroups(n);;for i in [1..len] do Print(Size(SmallGeneratingSet(SmallGroup(n,i))),","); od;
1,2,2,2,2,2,2,2,2,3,3,3,3,4,
gap> n:=6;;len:=NumberSmallGroups(n);;for i in [1..len] do Print(Size(SmallGeneratingSet(SmallGroup(n,i))),","); od;
2,1,
gap> n:=12;;len:=NumberSmallGroups(n);;for i in [1..len] do Print(Size(SmallGeneratingSet(SmallGroup(n,i))),","); od;
2,1,2,2,2,
gap> n:=24;;len:=NumberSmallGroups(n);;for i in [1..len] do Print(Size(SmallGeneratingSet(SmallGroup(n,i))),","); od;
2,1,2,2,2,2,2,2,2,2,2,2,2,3,3,
注意：n取其他值(例如n=6、12....)時GAP4.7/4.10.2使用Rank命令會報錯。
有限羣G嵌入S_n的最小n記爲m(G)=min{n∈N|G<S_n},則：m(C_2)=2,m(C_3)=3,m(C_4)=4,m(C_2×C_2)=4,問：試構造出來低階羣G的置換表示，並求出m(G)。
rank(D_4)=2,m(D_4)=4
rank(Q_8)=2,m(Q_8)<=8
rank(Q_12)=2,m(Q_12)<=12
rank(A_4)=2,m(A_4)=4
rank(D_6)=2,m(D_6)=6
羣的置換表示和GAP ID
3,9,12,13？
秩爲1
gap> IdGroup(Group([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)]));[ 16, 1 ]
秩爲2
gap> IdGroup(Group([(1,2,3,4),(5,6,7,8)]));[ 16, 2 ]
gap> IdGroup(Group((1,7,4,10),(5,6,8,9)));
[ 16, 2 ]
gap> IdGroup(Group((1,5)(2,6)(3,4,8,9)(7,10),(1,6)));
[ 16, 3 ]
兩個生成元分別是4階元和2階元
gap> IdGroup(Group([(1,2,3,4),(1,3)(5,6,7,8)]));[ 16, 4 ]
gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(9,10)]));[ 16, 5 ]
gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(1,5)(3,7)]));[ 16, 6 ]
gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(1,7)(3,5)(2,6)]));[ 16, 7 ]
gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(1,3)(5,7)(2,6)]));[ 16, 8 ]
秩爲3
gap> IdGroup(Group([(1,2,3,4),(5,6),(7,8)]));[ 16, 10 ]
gap> IdGroup(Group([(1,2,3,4),(1,3),(5,6)]));[ 16, 11 ]
秩爲4
gap> IdGroup(Group([(1,2),(3,4),(5,6),(7,8)]));[ 16, 14 ]
gap> IdGroup(ElementaryAbelianGroup(16));
[ 16, 14 ] 
F:\hxh1\MathTool\有限羣\G4
gap> IdGroup(Group((1,2,3,4)));
[ 4, 1 ]
gap> IdGroup(Group((3,4),(1,2)));
[ 4, 2 ]
gap> V4:=Group([(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)]);
Group([ (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ])
gap> IdGroup(V4);
[ 4, 2 ]
rank(S_3)=2,m(S_3)=3
rank(C_6)=1,m(C_6)=5
F:\hxh1\MathTool\有限羣\G6
gap> IdGroup(Group((2,4),(4,5)));
[ 6, 1 ]
gap> IdGroup(Group((1,2,3),(4,5)));
[ 6, 2 ]
gap> IdGroup(Group((1,2,3,4,5,6)));
[ 6, 2 ]
F:\hxh1\MathTool\有限羣\G8
rank(C_8)=1,m(C_8)<=8
gap> IdGroup(Group((1,2,3,4,5,6,7,8)));
[ 8, 1 ]
rank(C_4×C_2)=2,rank(C_4×C_2)=6
gap> IdGroup(Group((1,2),(3,4,5,6)));
[ 8, 2 ]
gap> IdGroup(Group((1,2,3,4),(5,6)));
[ 8, 2 ]
rank(D_4)=2,m(D_4)=4
gap> IdGroup(Group((1,2,3,4),(1,3)));
[ 8, 3 ]
rank(Q_8)=2,m(Q_8)<=8
gap> QuaternionGroup(IsPermGroup,8);
Group([ (1,5,3,7)(2,8,4,6), (1,2,3,4)(5,6,7,8) ])
rank(C_2×C_2×C_2)=3,m(C_8)=6
gap> IdGroup(Group((1,2),(3,4),(5,6)));
[ 8, 5 ]
F:\hxh1\MathTool\有限羣\G12
S_4、S_5不存在同構於[12,1]的子羣=>5<m(Q_12)<=12
m(C_12)=7
gap> IdGroup(Group((1,2,3,4,5,6)(7,8,9,10,11,12),(1,7,4,10)(2,12,5,9)(3,11,6,8)));
[ 12, 1 ]
gap> QuaternionGroup(IsPermGroup,12);
Group([ (1,7,4,10)(2,12,5,9)(3,11,6,8), (1,2,3,4,5,6)(7,8,9,10,11,12) ])
gap> IdGroup(Group((1,2,3),(4,5,6,7)));
[ 12, 2 ]
rank(A_4)=2,m(A_4)=4
gap> IdGroup(Group((1,2,3),(1,2)(3,4)));。
[ 12, 3 ]
gap> IdGroup(Group((),(1,2,3),(1,3,2),(1,2,4),(1,4,2),(1,3,4),(1,4,3),(2,3,4),(2,4,3),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)));
[ 12, 3 ]
rank(D_6)=2,m(D_6)=6
gap> IdGroup(Group((1,6)(4,5),(1,2,6,5,3,4)));
[ 12, 4 ]
gap> IdGroup(Group((1,3,5)(4,6),(1,5)));
[ 12, 4 ]
rank(C_2×C_2×C_3)=2,m(C_2×C_2×C_3)=7
gap> IdGroup(Group((1,2,3)(4,5),(4,5)(7,8)));
[ 12, 5 ]
gap> IdGroup(Group((1,2,3)(4,5)(6,7),(1,3,2)(6,7)));
[ 12, 5 ]