Leonard H. Soicher

Computing Galois groups over the rationals

Abstract Practical computational techniques are described to determine the Galois group of a polynomial over the rationals, and each transitive permutation group of degree 3 to 7 is realised as a Galois group over the rationals. The exact computations furnish a proof of the result.

Collection from the left and other strategies

We describe experiments with various collection strategies for the multiplication of elements of a p-group. We conclude that collection from the left is a very good strategy, then analyse the computational complexity of collection from the left, and compare it with that of collection from the right. Collection from the left also appears to be an excellent strategy for the multiplication of elements of a finite soluble group.

Low Rank Representations and Graphs for Sporadic Groups

1. Low rank permutation groups 2. Digraphs for transitive groups 3. The methods 4. The individual groups 5. Summary of the representations and graphs.

Symbolic Collection using Deep Thought

We describe the “Deep Thought” algorithm, which can, among other things, take a commutator presentation for a finitely generated torsion-free nilpotent group G , and produce explicit polynomials for the multiplication of elements of G . These polynomials were first shown to exist by Philip Hall, and allow for “symbolic collection” in finitely generated nilpotent groups. We discuss various practicalissues in calculations in such groups, including the construction of a hybrid collector, making use of both the polynomials and ordinary collection from the left.

Distance-transitive representations of the sporadic groups

A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free. In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.

An algorithmic approach to fundamental groups and covers of combinatorial cell complexes

Abstract We first develop a construction, originally due to Reidemeister, of the fundamental group and covers of a two-dimensional combinatorial cell complex. Then, we describe a practical algorithmic approach to the computation of fundamental groups and first homology groups (as finitely presented groups), of first homology groups mod p(as vector spaces), of deck groups (as permutation groups), and of covers of finite simple such complexes. In the case of clique complexes of finite simple graphs, the algorithms described have been implemented in GAP, making use of the GRAPE package.

Presentations of some 3-transposition groups

We classify all 3-transposition groups which are generated by at most five of their 3-transpositions. Modulo a center these are 27 specific groups plus various quotients of a particular group of order 2(349). For each of the 27 groups at least one presentation is given. We also give presentations for many groups of importance in the recent classification of 3-transposition groups with trivial center. Our presentations include ones for the three sporadic 3-transposition groups of Fischer, each on a 3-transposition generating set of minimal size.

More on block intersection polynomials and new applications to graphs and block designs

The concept of intersection numbers of order r for t-designs is generalized to graphs and to block designs which are not necessarily t-designs. These intersection numbers satisfy certain integer linear equations involving binomial coefficients, and information on the non-negative integer solutions to these equations can be obtained using the block intersection polynomials introduced by P.J. Cameron and the present author. The theory of block intersection polynomials is extended, and new applications of these polynomials to the studies of graphs and block designs are obtained. In particular, we obtain a new method of bounding the size of a clique in an edge-regular graph with given parameters, which can improve on the Hoffman bound when applicable, and a new method for studying the possibility of a graph with given vertex-degree sequence being an induced subgraph of a strongly regular graph with given parameters.

On balanced incomplete-block designs with repeated blocks

Balanced incomplete-block designs (BIBDs) with repeated blocks are studied and constructed. We continue work initiated by van Lint and Ryser in 1972 and pursued by van Lint in 1973. We concentrate on constructing (v,b,r,k,@l)-BIBDs with repeated blocks, especially those with gcd(b,r,@l)=1 and r@?20. We obtain new bounds for the multiplicity of a block in terms of the parameters of a BIBD, and improvements to these bounds for a resolvable BIBD. This allows us to answer a question of van Lint about the sufficiency of certain conditions for the existence of a BIBD with repeated blocks.

Computing with group homomorphisms

Let G= and H be finite groups and let o : X -> H be a map from the generating set X of G into H. We describe a simple approach for deciding whether or not o determines a group homomorphism from G to H, and if it does, for computing the kernel of o. If G and H are permutation groups the algorithm is a simple application of standard algorithms for bases and strong generating sets. If G and H are soluble groups given in the usual way by PAG-systems and corresponding power conjugate presentations then the algorithm is a simple application of the non-commutative Gauss algorithm for constructing a subgroup of a soluble group. Further, a probabilistic algorithm is given for finding the kernel and image of o when each of G and H is given as a permutation group or a soluble group, |G| is known, and o is known to determine a homomorphism.