Greg Gamble - Academia.edu (original) (raw)

Papers by Greg Gamble

Research paper thumbnail of ACE: A GAP 4 Package Providing an Interface to the Advanced Coset Enumerator

World Wide Web - WWW, 2003

Research paper thumbnail of 1.5 Using ACE Interactively.... 10

Research paper thumbnail of Classification of partitions of all triples on ten points into copies of Fano and affine planes

Discrete Mathematics, 2008

We continue our study of partitions of the full set of v 3 triples chosen from a v-set into copie... more We continue our study of partitions of the full set of v 3 triples chosen from a v-set into copies of the Fano plane PG(2, 2) (Fano partitions) or copies of the affine plane AG(2, 3) (affine partitions) or into copies of both of these planes (mixed partitions). The smallest cases for which such partitions can occur are v = 8 where Fano partitions exist, v = 9 where affine partitions exist, and v = 10 where both affine and mixed partitions exist. The Fano partitions for v = 8 and the affine partitions for v = 9 and 10 have been fully classified, into 11, two and 77 isomorphism classes, respectively. Here we classify (1) the sets of i pairwise disjoint affine planes for i = 1,. .. , 7, and (2) the mixed partitions for v = 10 into their 22 isomorphism classes. We consider the ways in which these partitions relate to the large sets of AG(2, 3).

Research paper thumbnail of BODMAS, BOMDAS and DAMNUS ... the Sequel

Research paper thumbnail of A GAP4 program found in the ACE package

Research paper thumbnail of Block-transitive 3-designs with affine automorphism group

Journal of Geometry, 1994

Let X = (Z p) d where p is an odd prime and d ∈ N, and let B ⊆ X, |B| = k. Then it was shown by P... more Let X = (Z p) d where p is an odd prime and d ∈ N, and let B ⊆ X, |B| = k. Then it was shown by Praeger that the set B = {B g | g ∈ AGL d (p)} is the block-set of a 3-design if and only if the number q 1 (k) of collinear triples of points of B is q 1 (k) = k(k − 1)(k − 2)(p − 2) 6(p d − 2). We give an explicit method for determining all k such that 3 ≤ k < p d /2 and q 1 (k) is an integer. For p = 3 the smallest value of d for which there is such an integral value of q 1 (k) is d = 7 and the smallest k ≥ 3 giving integral values of q 1 (k) are k = 115 and k = 116. We construct many examples of 3-designs (X, B) with k = 115 or 116 admitting AGL 7 (3) as automorphism group.

Research paper thumbnail of Symmetric Difference-Free and Symmetric Difference-Closed Collections of Sets

Graphs and Combinatorics, 2013

ABSTRACT A collection of sets is symmetric-difference-free, respectively symmetric difference-clo... more ABSTRACT A collection of sets is symmetric-difference-free, respectively symmetric difference-closed, if the symmetric difference of any two sets in the collection lies outside, respectively inside, the collection. Recently Buck and Godbole (Size-maximal symmetric difference-free families of subsets of [n], Graphs Combin. (to appear), 2013) investigated such collections and showed, in particular, that the the largest symmetric difference-free collection of subsets of an n-set has cardinality 2 n-1. We use group theory to obtain shorter proofs of their results.

Research paper thumbnail of Further results on strongbox secured secret sharing schemes

Research paper thumbnail of Vertex-primitive groups and graphs of order twice the product of two distinct odd primes

Journal of Group Theory, 2000

A non-Cayley number is an integer n for which there exists a vertex-transitive graph on n vertice... more A non-Cayley number is an integer n for which there exists a vertex-transitive graph on n vertices which is not a Cayley graph. In this paper, we complete the determination of the non-Cayley numbers of the form 2 pq, where pY q are distinct odd primes. Earlier work of Miller and the second author had dealt with all such numbers corresponding to vertex-transitive graphs admitting an imprimitive subgroup of automorphisms. This paper deals with the primitive case. First the primitive permutation groups of degree 2 pq are classi®ed. This depends on the ®nite simple group classi®cation. Then each of these groups G is examined to determine whether there are any non-Cayley graphs which admit G as a vertex-primitive subgroup of automorphisms, and admit no imprimitive subgroups. The outcome is that 2 pq is a non-Cayley number, where 2`q`p and q and p are primes, if and only if one of pI1 mod 4, or qI1 mod 4, or pI1 mod q, or p 4q À 1, or pY q 11Y 7 or 19Y 7 holds.

Research paper thumbnail of ACE: A GAP 4 Package Providing an Interface to the Advanced Coset Enumerator

World Wide Web - WWW, 2003

Research paper thumbnail of 1.5 Using ACE Interactively.... 10

Research paper thumbnail of Classification of partitions of all triples on ten points into copies of Fano and affine planes

Discrete Mathematics, 2008

We continue our study of partitions of the full set of v 3 triples chosen from a v-set into copie... more We continue our study of partitions of the full set of v 3 triples chosen from a v-set into copies of the Fano plane PG(2, 2) (Fano partitions) or copies of the affine plane AG(2, 3) (affine partitions) or into copies of both of these planes (mixed partitions). The smallest cases for which such partitions can occur are v = 8 where Fano partitions exist, v = 9 where affine partitions exist, and v = 10 where both affine and mixed partitions exist. The Fano partitions for v = 8 and the affine partitions for v = 9 and 10 have been fully classified, into 11, two and 77 isomorphism classes, respectively. Here we classify (1) the sets of i pairwise disjoint affine planes for i = 1,. .. , 7, and (2) the mixed partitions for v = 10 into their 22 isomorphism classes. We consider the ways in which these partitions relate to the large sets of AG(2, 3).

Research paper thumbnail of BODMAS, BOMDAS and DAMNUS ... the Sequel

Research paper thumbnail of A GAP4 program found in the ACE package

Research paper thumbnail of Block-transitive 3-designs with affine automorphism group

Journal of Geometry, 1994

Let X = (Z p) d where p is an odd prime and d ∈ N, and let B ⊆ X, |B| = k. Then it was shown by P... more Let X = (Z p) d where p is an odd prime and d ∈ N, and let B ⊆ X, |B| = k. Then it was shown by Praeger that the set B = {B g | g ∈ AGL d (p)} is the block-set of a 3-design if and only if the number q 1 (k) of collinear triples of points of B is q 1 (k) = k(k − 1)(k − 2)(p − 2) 6(p d − 2). We give an explicit method for determining all k such that 3 ≤ k < p d /2 and q 1 (k) is an integer. For p = 3 the smallest value of d for which there is such an integral value of q 1 (k) is d = 7 and the smallest k ≥ 3 giving integral values of q 1 (k) are k = 115 and k = 116. We construct many examples of 3-designs (X, B) with k = 115 or 116 admitting AGL 7 (3) as automorphism group.

Research paper thumbnail of Symmetric Difference-Free and Symmetric Difference-Closed Collections of Sets

Graphs and Combinatorics, 2013

ABSTRACT A collection of sets is symmetric-difference-free, respectively symmetric difference-clo... more ABSTRACT A collection of sets is symmetric-difference-free, respectively symmetric difference-closed, if the symmetric difference of any two sets in the collection lies outside, respectively inside, the collection. Recently Buck and Godbole (Size-maximal symmetric difference-free families of subsets of [n], Graphs Combin. (to appear), 2013) investigated such collections and showed, in particular, that the the largest symmetric difference-free collection of subsets of an n-set has cardinality 2 n-1. We use group theory to obtain shorter proofs of their results.

Research paper thumbnail of Further results on strongbox secured secret sharing schemes

Research paper thumbnail of Vertex-primitive groups and graphs of order twice the product of two distinct odd primes

Journal of Group Theory, 2000

A non-Cayley number is an integer n for which there exists a vertex-transitive graph on n vertice... more A non-Cayley number is an integer n for which there exists a vertex-transitive graph on n vertices which is not a Cayley graph. In this paper, we complete the determination of the non-Cayley numbers of the form 2 pq, where pY q are distinct odd primes. Earlier work of Miller and the second author had dealt with all such numbers corresponding to vertex-transitive graphs admitting an imprimitive subgroup of automorphisms. This paper deals with the primitive case. First the primitive permutation groups of degree 2 pq are classi®ed. This depends on the ®nite simple group classi®cation. Then each of these groups G is examined to determine whether there are any non-Cayley graphs which admit G as a vertex-primitive subgroup of automorphisms, and admit no imprimitive subgroups. The outcome is that 2 pq is a non-Cayley number, where 2`q`p and q and p are primes, if and only if one of pI1 mod 4, or qI1 mod 4, or pI1 mod q, or p 4q À 1, or pY q 11Y 7 or 19Y 7 holds.