The function DifferenceSet given below may be used to construct the design corresponding to a given set S of integers forming a perfect difference set modulo n.
> DifferenceSet := func< S, n |
> [ { (x + i) mod n + 1 : x in S } : i in [0..n-1] ] >;
We use this function to construct a (19, 9, 4)-design
and then proceed to compute its automorphism group.
> DSet := {1, 4, 5, 6, 7, 9, 11, 16, 17};
> D := Design< 2, 19 | DifferenceSet(DSet, 19) >;
> D;
2-(19, 9, 4) Design with 19 blocks
> Blocks(D);
{@
{2, 5, 6, 7, 8, 10, 12, 17, 18}, {3, 6, 7, 8, 9, 11, 13, 18, 19},
{1, 4, 7, 8, 9, 10, 12, 14, 19}, {1, 2, 5, 8, 9, 10, 11, 13, 15},
{2, 3, 6, 9, 10, 11, 12, 14, 16}, {3, 4, 7, 10, 11, 12, 13, 15, 17},
{4, 5, 8, 11, 12, 13, 14, 16, 18}, {5, 6, 9, 12, 13, 14, 15, 17, 19},
{1, 6, 7, 10, 13, 14, 15, 16, 18}, {2, 7, 8, 11, 14, 15, 16, 17, 19},
{1, 3, 8, 9, 12, 15, 16, 17, 18}, {2, 4, 9, 10, 13, 16, 17, 18, 19},
{1, 3, 5, 10, 11, 14, 17, 18, 19}, {1, 2, 4, 6, 11, 12, 15, 18, 19},
{1, 2, 3, 5, 7, 12, 13, 16, 19}, {1, 2, 3, 4, 6, 8, 13, 14, 17},
{2, 3, 4, 5, 7, 9, 14, 15, 18}, {3, 4, 5, 6, 8, 10, 15, 16, 19},
{1, 4, 5, 6, 7, 9, 11, 16, 17}
@}
> Aut := AutomorphismGroup(D);
> A := OrbitImage(Aut, 1);
> A;
Permutation group A acting on a set of cardinality 19
Order = 171 = 3^2 * 19
(1, 4, 12, 8, 10, 9, 19, 14, 7)(2, 13, 17, 15, 16, 6, 11, 18, 5)
(2, 5, 17, 8, 10, 18, 12, 7, 6)(3, 9, 14, 15, 19, 16, 4, 13, 11)
> IsPrimitive(A);
true
> IsFrobenius(A);
true
Thus the group is a Frobenius group and the normal subgroup of order 19
must be the Frobenius kernel.
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