Donald E. Taylor

Computing in groups of Lie type

We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction.

On outer automorphism groups of coxeter groups

SummaryIt is shown that the outer automorphism group of a Coxeter groupW of finite rank is finite if the Coxeter graph contains no infinite bonds. A key step in the proof is to show that if the group is irreducible andΠ1 andΠ2 any two bases of the root system ofW, thenΠ2 = ±ωΠ1 for some ω εW. The proof of this latter fact employs some properties of the dominance order on the root system introduced by Brink and Howlett.

Vertex-primitive half-transitive graphs

Given an infinite family of finite primitive groups, conditions are found which ensure that almost all the orbitals are not self-paired. If p is a prime number congruent to ±1(mod 10), these conditions apply to the groups P S L (2, p ) acting on the cosets of a subgroup isomorphic to A 5 . In this way, infinitely many vertex-primitive ½-transitive graphs which are not metacirculants are obtained.

Two-graph and doubly transitive groups

Abstract Using the classification of finite doubly transitive groups the finite two-graphs which admit a doubly transitive group of automorphisms are determined.

Matrix Generators for Exceptional Groups of Lie Type

This paper gives a uniform method of constructing generators for matrix representations of finite groups of Lie type with particular emphasis on the exceptional groups. The algorithm constructs matrices for the action of root elements on the lowest dimension representation of an associated Lie algebra. These generators have been implemented in the computer algebra system Magma and this completes the provision of pairs of matrix generators for all finite groups of Lie type.

Polynomial-time versions of Sylow's theorem

Abstract Let G be a subgroup of Sn, given in terms of a generating set of permutations, and let p be a prime divisor of |G|. If G is solvable—and, more generally, if the nonabelian composition factors of G are suitably restricted—it is shown that the following can be found in polynomial time: a Sylow p-subgroup of G containing a given p-subgroup, and an element of G conjugating a given Sylow p-subgroup to another. Similar results are proved for Hall subgroups of solvable groups and a version of the Schur-Zassenhaus theorem is obtained.

A Group-Theoretic Method for Drawing Graphs Symmetrically

Constructing symmetric drawings of graphs is NP-hard. In this paper, we present a new method for drawing graphs symmetrically based on group theory. More formally, we define a n-geometric automorphism group of a graph that can be displayed as symmetries of a drawing of the graph in n dimensions. Then we present an algorithm to find all 2- and 3-geometric automorphism groups of a graph. We implement the algorithm using Magma [11] and the experimental results shows that our approach is very efficient in practice. We also present a drawing algorithm to display a 2- or 3-geometric automorphism group.

Matrix Generators for the Orthogonal Groups

In 1962 Steinberg gave pairs of generators for all finite simple groups of Lie type. In this paper, for each finite orthogonal group we provide a pair of matrices which generate its derived group: the matrices correspond to Steinbergs generators modulo the centre. These generators have been implemented in the computer algebra system MAGMA and this completes the provision of pairs of generators in MAGMA for all (perfect) finite classical groups.

Geometric automorphism groups of graphs

Constructing symmetric drawings of graphs is NP-hard. In this paper, we present a new method for drawing graphs symmetrically based on group theory. More formally, we define an n-geometric automorphism group as a subgroup of the automorphism group of a graph that can be displayed as symmetries of a drawing of the graph in n dimensions. Then we present an algorithm to find all 2- and 3-geometric automorphism groups of a given graph. We implement the algorithm using Magma [] and the experimental results show that our approach is very efficient in practice. We also present a drawing algorithm to display 2- and 3-geometric automorphism groups.

On a Certain Lie Algebra Defined by a Finite Group

Some years ago W. Plesken told the first author of a simple but interesting construction of a Lie algebra from a finite group. The authors posed themselves the question as to what the structure of this Lie algebra might be. In particular, for which groups does the construction produce a simple Lie algebra? The answer is given in the present paper; it uses some textbook results on representations of finite groups, which we explain along the way. Little knowledge of the theory of Lie algebras is required beyond the dfinition of a Lie algebra itself and the definitions of simple and semisimple Lie algebras. Thus this exposition may serve as the basis for some entertaining examples er exercises in a graduate course on the representation theory of finite groups.