Construct the automorphism group of the incidence structure D. If the incidence structure D is simple, the automorphism group acts on the point set of D. Otherwise, it acts on the union of the point set and the block set.
Construct the automorphism group of the incidence structure D in its action on the point set of D.
Construct the automorphism group of the incidence structure D in its action on the block set of D.
The image of the point u of the incidence structure D under the action of the automorphism a of D (returns a point).
The image of the set S of points of the incidence structure D under the action of the automorphism a of D (returns a set of points).
The image of the sequence S of points of the incidence structure D under the action of the automorphism a of D (returns a sequence of points).
The image of the block b of the incidence structure D under the action of the automorphism a of D (returns a block).
The image of the set T of blocks of the incidence structure D under the action of the automorphism a of D (returns a set of blocks).
The image of the sequence U of blocks of the incidence structure D under the action of the automorphism a of D (returns a sequence of blocks).
The image of the point u of the incidence structure D under the action of the subgroup B of the automorphism group of D (returns a set of points).
The image of the set S of points of the incidence structure D under the action of the subgroup B of the automorphism group of D (returns a set of sets of points).
The image of the sequence Q of blocks of the incidence structure D under the action of the subgroup B of the automorphism group of D (returns a set of sequences of blocks).
The image of the block b of incidence structure D under the action of the subgroup B of the automorphism group of D (returns a set of blocks).
The image of the set T of blocks of the incidence structure D under the action of the subgroup B of the automorphism group of D (returns a set of sets of blocks).
The image of the sequence U of blocks of the incidence structure D under the action of the subgroup B of the automorphism group of D (returns a set of sequences of blocks).
Given an incidence structure D on a set of n points and a permutation a of Sym(n), construct the image of the incidence structure D under a (returns an incidence structure).
Given an incidence structure D on a set of n points and a subgroup A of Sym(n), construct the orbit of the incidence structure D under A (returns a set of incidence structures).
Given a point u belonging to the point set of the incidence structure D, construct the stabilizer of u in the subgroup A of Aut(D).
Given a sequence Q of points belonging to the point set of the incidence structure D, construct the pointwise stabilizer of Q in the subgroup A of Aut(D).
Given a set S of points belonging to the point set of the incidence structure D, construct the stabilizer of S in the subgroup A of Aut(D).
Given a block b belonging to the block set of the incidence structure D, construct the stabilizer of b in the subgroup A of Aut(D).
Given a sequence Q of blocks belonging to the block set of the incidence structure D, construct the pointwise stabilizer of Q in the subgroup A of Aut(D).
IsPointTransitive(D) : Inc -> BoolElt
True iff the automorphism group of the incidence structure D acts transitively on the point set of D.
True iff the automorphism group of the incidence structure D acts transitively on the block set of D.
> D, P, B := WittDesign(12);
> A := AutomorphismGroup(D);
> print A;
Permutation group A acting on a set of cardinality 12
Order = 95040 = 2^6 * 3^3 * 5 * 11
(1, 2)(5, 8)(6, 11)(10, 12)
(2, 3)(5, 12)(6, 8)(10, 11)
(3, 4)(5, 10)(6, 11)(8, 12)
(4, 7)(5, 8)(6, 12)(10, 11)
(5, 11, 12, 6)(7, 8, 9, 10)
(5, 9, 12, 7)(6, 10, 11, 8)
(5, 12)(6, 11)(7, 9)(8, 10)
> a := Random(A);
> print a;
(3, 9)(4, 10, 8, 6, 7, 11, 5, 12)
> print 4^a;
10
> print {1, 2, 3}^a;
{ 1, 2, 9 }
> print Block(D, 3)^a;
{1, 2, 3, 4, 8, 10}
> Asub := sub< A | A.1, A.2 >;
> print Asub;
Permutation group Asub acting on a set of cardinality 12
(1, 2)(5, 8)(6, 11)(10, 12)
(2, 3)(5, 12)(6, 8)(10, 11)
> print Stabilizer(Asub, Block(D, 3));
Permutation group acting on a set of cardinality 12
Order = 2
(1, 2)(5, 8)(6, 11)(10, 12)
> print Stabilizer(Asub, 6);
Permutation group acting on a set of cardinality 12
Order = 1
Id($)
> print 3^Asub;
GSet{ 1, 2, 3 }
> print IsPointTransitive(D);
true
The design D has a resolution; i.e. the block set of D can be
partitioned into parallel classes. In this example, the automorphism
group A acts transitively on the classes of the resolution. Hence
we can get the resolution of D by taking the orbit of a single
parallel class under A.
> print D;
5-(12, 6, 1) Design with 132 blocks
> b1 := B.1;
> b2 := B ! (Points(D) diff b1);
> class := {b1, b2};
> print class;
{ {1, 6, 7, 8, 11, 12}, {2, 3, 4, 5, 9, 10} }
> resolution := class^A;
> print #resolution;
66