Bettina Eick

A MILLENNIUM PROJECT: CONSTRUCTING SMALL GROUPS

We survey the problem of constructing the groups of a given finite order. We provide an extensive bibliography and outline practical algorithmic solutions to the problem. Motivated by the millennium, we used these methods to construct the groups of order at most 2000; we report on this calculation and describe the resulting group library.

CONSTRUCTING AUTOMORPHISM GROUPS OF p-GROUPS

ABSTRACT We present an algorithm to construct the automorphism group of a finite p-group. The method works down the lower exponent-p central series of the group. The central difficulty in each inductive step is a stabiliser computation; we introduce various approaches designed to simplify this computation.

Construction of Finite Groups

We1999 Academic Pressintroduce three practical algorithms to construct certain finite groups up to isomorphism. The first one can be used to construct all soluble groups of a given order. This method can be restricted to compute the soluble groups with certain properties such as nilpotent, non-nilpotent or supersoluble groups. The second algorithm can be used to determine the groups of orderCopyright pn· qwith a normal Sylow subgroup for distinct primespandq. The third method is a general method to construct finite groups which we use to compute insoluble groups.

The groups of order at most 2000

We announce the construction up to isomorphism of the 49 910 529 484 groups of order at most 2000.

The Groups of Order at Most 1000 Except 512 and 768

Recently, we developed practical algorithms to determine up to isomorphism the groups of a given order. Here we describe details on the implementations and the applications of these methods. In particular, we report on the determination of the groups of order at most 1000 except 512 and 768.

Computing polycyclic presentations for polycyclic rational matrix groups

We describe practical algorithms for computing a polycyclic presentation and for facilitating a membership test for a polycyclic subgroup of GL(d,Q). A variation of this method can be used to check whether a finitely generated subgroup of GL(d,Q) is solvable or solvable-by-finite. We report on our implementations of the algorithms for determining a polycyclic presentation and checking solvability.

On the orbit-stabilizer problem for integral matrix actions of polycyclic groups

We present an algorithm to solve the orbit-stabilizer problem for a polycyclic group G acting as a subgroup of GL(d, Z) on the elements of Qd. We report on an implementation of our method and use this to observe that the algorithm is practical.

Some new simple Lie algebras in characteristic 2

We describe an algorithm for computing automorphism groups and testing isomorphisms of finite dimensional Lie algebras over finite fields. The algorithm is particularly effective for simple or almost simple Lie algebras. We show how it can be used in a computer search for new low dimensional simple Lie algebras over the field with two elements.

COMPUTATION OF LOW-DIMENSIONAL (CO)HOMOLOGY GROUPS FOR INFINITE SEQUENCES OF p-GROUPS WITH FIXED COCLASS

Eick and Leedham-Green introduced infinite sequences of p-groups of fixed coclass. Here we describe an algorithm to compute the Schur multiplicators of all groups in an infinite sequence simultaneously. Based on this, we prove that these Schur multiplicators can be described by a single parametrised presentation; this confirms a conjecture by Eick. Similar results are obtained for certain low-dimensional cohomology groups; these results support a conjecture by Carlson.

A NILPOTENT QUOTIENT ALGORITHM FOR CERTAIN INFINITELY PRESENTED GROUPS AND ITS APPLICATIONS

The main part of this paper contains a description of a nilpotent quotient algorithm for L-presented groups and a report on applications of its implementation in the computer algebra system GAP. The appendix introduces two new infinite series of L-presented groups. Apart from being of interest in their own right, these new L-presented groups serve as examples for applications of the nilpotent quotient algorithm.We describe a nilpotent quotient algorithm for a certain class of infinite presentations: the so-called finite L-presentations. We then exhibit finite L-presentations for various examples and report on the application of our nilpotent quotient algorithm to them. As a result, we obtain conjectural descriptions of the lower central series structure of various interesting groups including the Grigorchuk supergroup, the Brunner–Sidki–Vieira group, the Basilica group, certain generalizations of the Fabrykowski–Gupta group, and certain generalizations of the Gupta–Sidki group.