We construct the automorphism group of the block design corresponding to the projective plane .
> B1 := { 1, 2, 4 };
> B2 := { 2, 3, 5 };
> B3 := { 3, 4, 6 };
> B4 := { 4, 5, 7 };
> B5 := { 5, 6, 1 };
> B6 := { 6, 7, 2 };
> B7 := { 7, 1, 3 };
> D := Design< 2, 7 | B1, B2, B3, B4, B5, B6, B7 >;
> D;
2-(7, 3, 1) Design with 7 blocks
> A, PP, BB := AutomorphismGroup(D);
> A;
Permutation group A acting on a set of cardinality 7
Order = 168 = 2^3 * 3 * 7
(1, 2)(3, 5, 6, 7)
(2, 5, 3)(4, 6, 7)
(2, 4)(3, 5, 7, 6)
(3, 6)(5, 7)
(3, 5)(6, 7)
> PP;
GSet{ 1, 2, 3, 4, 5, 6, 7 }
> BB;
GSet{ {1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 7}, {1, 5, 6},
{2, 6, 7}, {1, 3, 7} }
The returned values PP and BB are A-sets corresponding to the
action of the automophism group on the points and blocks of D
respectively. Using them, we can determine the stabilizer of the
block B1 in two ways--first, by regarding B1 as a set of points
to be stabilized under the action of A on the points; and second,
by regarding B1 as a block to be fixed under the action of A on the
blocks.
> StabB1_1 := Stabilizer(A, PP, B1);
> StabB1_2 := Stabilizer(A, BB, B1);
> StabB1_1;
Permutation group StabB1_1 acting on a set of cardinality 7
Order = 24 = 2^3 * 3
(3, 7)(5, 6)
(2, 4)(5, 6)
(1, 2)(5, 7)
> StabB1_1 eq StabB1_2;
true
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