To save space and avoid clutter the examples are shown without the gap>
and > prompts, as they might appear in an input file. For examples of
Example of Double Coset Enumeration Strategies.
The Symmetric group of degree 5
It is well known that
G = S_5 = leftlangle,a,b,c,d right.&mid& a^2 = b^2 = c^2 = d^2 =
(ab)^3 = (bc)^3 = (cd)^3 =
&&left.(ac)^2 = (ad)^2 = (bd)^2 = 1rightrangle.
For brevity we denote this presentation by a Coxeter diagram:
setlengthunitlength1cm put(0.5,0.5)line(1,0)3 multiput(0.5,0.5)(1,0)4circle0.1 put(0.5,0.6)makebox(0,0)[b]a put(1.5,0.6)makebox(0,0)[b]b put(2.5,0.6)makebox(0,0)[b]c put(3.5,0.6)makebox(0,0)[b]d
We let K=gen(a,b)iso S_3 and identify a with (1,2) and b with (2,3). Then G is generated by K and X = {c,d}. We can see from the presentation that L_c = gen(a) = Stab_K(3), while L_d = K. We set H=gen(a,b,c)iso S_4 and obtain the following presentation:
gap> k := SymmetricGroup(3);
Group( (1,3), (2,3) )
gap> c := AbstractGenerator("c");;
gap> d := AbstractGenerator("d");;
gap> S5Pres := rec(
> groupK := k,
> gainGroups := [rec(), # default to L=K
> rec(dom := 3)], # default to action on points
> gens := [rec(name := c, invol := true, wgg := 2),
> rec(name := d, invol := true, wgg := 1)],
> relators := [DCEWord(k,c*d)^3,DCEWord(k,[(2,3),c])^3],
> subgens := [DCEWord(k,(1,2,3)), DCEWord(k,(1,2)), DCEWord(k,c)]);;
The Weyl Group of Type E_6
We consider another group given by a Coxeter presentation
put(0.5,0.5)line(1,0)4 multiput(0.5,0.5)(1,0)5circle0.1 put(0.5,0.6)makebox(0,0)[b]a put(1.5,0.6)makebox(0,0)[b]b put(2.5,0.6)makebox(0,0)[b]c put(3.5,0.6)makebox(0,0)[b]d put(4.5,0.6)makebox(0,0)[b]e put(2.5,-0.5)circle0.1 put(2.5,-0.5)line(0,1)1 put(2.6,-0.5)makebox(0,0)[l]f
This time we take K=H=gen(a,b,c,d,e)iso S_6 and obtain the presentation:
gap> k := SymmetricGroup(6);
Group( (1,6), (2,6), (3,6), (4,6), (5,6) )
gap> f := AbstractGenerator("f");;
gap> WE6Pres := rec (
> groupK := k,
> gainGroups := [rec(dom := [1,2,3])], # action defaults to OnSets
> gens := [rec(name := f,wgg := 1, invol := true)],
> relators := [DCEWord(k,[(3,4),f])^3],
> subgens := [DCEWord(k,(1,2,3,4,5,6)),DCEWord(k,(1,2))] );;
S_5 revisited
To illustrate other features of the program, we consider the presentation
of S_5 again, but this time we choose K=gen(b,c)iso S_3 and
identify b with (1,2) and c with (2,3). Now L_a = gen(c) =
Stab_K(1), while L_d = gen(b) = Stab_K(3). These are two conjugate
subgroups of K, so we can use the ggconj feature to combine the data
structures for them.
We can present this as:
gap> k := SymmetricGroup(3);
Group( (1,3), (2,3) )
gap> a := AbstractGenerator("a");;
gap> d := AbstractGenerator("d");;
gap> b := (1,2);;
gap> c := (2,3);;
gap> S5PresA := rec (
> groupK := k,
> gainGroups := [rec(dom := 1)],
> gens := [rec(name := a, invol := true, wgg := 1),
> rec(name := d, invol := true, wgg := 1, ggconj := (1,3))],
> relators := [DCEWord(k,[c,d])^3,DCEWord(k,[a,b])^3,
> DCEWord(k,[a,d])^2],
> subgens := [DCEWord(k,(1,2,3)),DCEWord(k,(1,2)), DCEWord(k,a)] );;
The Harada-Norton Group
The almost-simple group HN: 2 can be constructed as follows. Take the symmetric group S_{12} acting naturally on {1,ldots,12} and let L be the stabiliser of {1,2,3,4,5,6}. Then L iso S_6times S_6. Extend S_{12} by adjoining an element a which normalizes L and acts on each factor S_6 by its outer automorphism. Impose the additional relations a^2 = 1 and (a(6,7))^5=1.
With H=K, this construction translates directly into DCE input:
gap> a := AbstractGenerator("a");;
gap> K := Group((1,2,3,4,5,6,7,8,9,10,11,12),(1,2));;
gap> L := Stabilizer(K,[1,2,3,4,5,6],OnSets);;
gap> f := GroupHomomorphismByImages( L,L,
> [(1,5,4,3,2),(5,6),(12,8,9,10,11),(7,8)],
> [(1,5,4,3,2),(1,4)(2,3)(5,6),(12,8,9,10,11),(12,9)(10,11)(7,8)] );;
gap> HNPres := rec(
> groupK := K,
> gainGroups := [ rec(dom := [1,2,3,4,5,6], op := OnSets)],
> gens := [ rec( name := a, invol := true, wgg := 1, action := f)],
> relators := [DCEWord(K,[a,(6,7)])^5],
> strategy := rec(whichStrategy := "HLT", EC := [1140000]),
> subgens := [(1,2,3,4,5,6,7,8,9,10,11,12),(1,2)]);;
A Non-permutation Example
The programs were written with the case of K a permutation group uppermost in the author's mind, however other representations are possible.
In this example, we represent the symmetric group S_4 as an AG-group in the Coxeter presentation of S_6. This example also demonstrates the explicit use of the action of K on left cosets of L_x, when no suitable action on points, sets or similar is available.
gap> k := AgGroup(SymmetricGroup(4));
Group( g1, g2, g3, g4 )
gap> a := PreImage(k.bijection,(1,2));
g1
gap> b := PreImage(k.bijection,(2,3));
g1*g2
gap> c := PreImage(k.bijection,(3,4));
g1*g4
gap> d := AbstractGenerator("d");;
gap> e := AbstractGenerator("e");;
gap> l := Subgroup(k,[a,b]);
Subgroup( Group( g1, g2, g3, g4 ), [ g1, g1*g2 ] )
gap> OurOp := function(cos,g)
> return g^-1*cos; # note the inversion
> end;;
gap> Pres := rec (
> groupK := k,
> gainGroups := [rec(dom := k.identity*l, op := OurOp),rec()],
> gens := [rec(name := d, invol := true, wgg := 1),
> rec(name := e, invol := true, wgg := 2)],
> relators := [DCEWord(k,d*e)^3,DCEWord(k,[c,d])^3],
> subgens := [DCEWord(k,a),DCEWord(k,b), DCEWord(k,c),DCEWord(k,d)]);;
GAP 3.4.4