Charles C. Sims

A simple group of order 44,352,000

The group G of the title is obtained as a primitive permutation group of degree 100 in which the stabilizer of a point has orbits of lengths 1, 22 and 77 and is isomorphic to the Mathieu group M22. Thus G has rank 3 in the sense of [1]. G is an automorphism group of a graph constructed from the Steiner system ~ (3, 6, 22). WITT [3] defined a Steiner system ~(d, rn, n) to be a set S of n points together with a set B of subsets of S (referred to here as blocks) such that each block contains exactly m points and each set of d points is contained in exactly one block. WITT [4] showed that Steiner systems ~ (3, 6, 22) exist and that they are unique up to isomorphism. The automorphism group Mz2 of an ~ (3, 6, 22) contains the Mathieu group Mz2 as a subgroup of index 2 and is the normalizer of M22 in M24. Throughout the rest of the paper we shall use the following notation: S and B will denote the sets of points and blocks, respectively, of a fixed ~(3, 6, 22). Points will be denoted by Greek letters ~, fl, ... and blocks by Roman letters u, v, .... For each o~eS, [~] will denote the set of blocks containing ~. We shall use the following facts about ~(3, 6, 22) and M22:

Verifying nilpotence

This paper describes a new procedure, based on string rewriting rules, for verifying that a finitely presented group G is nilpotent. If G is not nilpotent, the procedure may not terminate. A preliminary computer implementation of the procedure has been used to prove a theorem about minimal presentations of free nilpotent groups of class 3. Finally, it is shown that the ideas presented here may be combined with work of Baumslag et al. (1981) to prove that the polycyclicity of a finitely presented group can be verified.

A polycyclic quotient algorithm

This paper describes a generalization of the Grobner basis method to the integral group ring of a polycyclic group. A polycyclic quotient algorithm is developed using this method. SupposeGis a group given by a finite presentation andG(n)is thenth term in the derived series ofG. A polycyclic quotient algorithm computes the quotientG/G(n)if it is polycyclic. An implementation of this algorithm in C has been developed and its efficiency is encouraging.

Computing the order of a solvablepermutation group

This paper presents a new algorithm for determining the order of a solvable permutationgroup from a set of generators. During the computation, a base and strong generating set and a polycyclic generating sequence are also obtained.

A presentation for the Lyons simple group

We give a presentation of the Lyons simple group together with information on a complete computational proof that the presentation is correct. This fills a longstanding gap in the literature on the sporadic simple groups. This presentation is a basis for various matrix and permutation representations of the group.

The Knuth-Bendix procedure for strings as a substitute for coset enumeration

This note describes two examples in which the Knuth-Bendix procedure for strings is more useful than coset enumeration for studying a finitely presented group.

Implementing the Baumslag-Cannonito-Miller polycyclic quotient algorithm

This paper discusses the practical problems associated with developing a computerimplementation of an algorithm described by Baumslag, Cannonito, and Miller for computing polycyclic quotients of a finitely presented group. Attention is drawn to the connection with the method of Grobner bases and techniques for computing nilpotent quotients. Some experimentation with computing class-2 nilpotent quotients and metabelian polycyclic quotients is described.

Groups with exponent six

Burnside asked questions about periodic groups in his influential paper of 1902. The study of groups with exponent six is a special case of the study of the Burnside questions on which there has been significant progress. It has contributed a number of worthwhile aspects to the theory of groups and in particular to computation related to groups. Finitely generated groups with exponent six are finite. We investigate the nature of relations required to provide proofs of finiteness for some groups with exponent six. We give upper and lower bounds for the number of sixth powers needed to define the largest 2-generator group with exponent six. We solve related questions about other groups with exponent sis using substantial computations which we explain.

Fast multiplication and growth in groups

To examine computationally the growth rate of a group with respect to a set of generators, one needs to be able to multiply elements of the group very quickly and to store large sets of group elements. The paper describes methods for addressing both of these issues in three classes of groups: nite groups of prime exponent, free nilpotent groups of nite rank, and nite groups with subgroup chains having moderate-sized indexes. These techniques have made it possible to explore balls about the identity with roughly 10

Computing with subgroups of automorphism groups of finite groups

Let G be a finite group which possesses a chain of characteristic subgroups with quotients of moderate size. This paper describes methods for computing the order of and for deciding membership in a subgroup of the automorphism group of G. An application to the study of the Burnside group i3(2, 5) is discussed.