E. A. O'Brien

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.

The p-group generation algorithm

The theory and implementation of an algorithm used in generating descriptions of p-groups aredescribed. Some applications and details of the performance of the algorithm are provided.

The groups of order 256

Abstract Building on earlier work, a new method for generating descriptions of p -groups is developed. The theory and implementation of this method are described and its application in determining the 56 092 groups of order 256 is outlined.

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.

Isomorphism testing for p -groups

We describe the theoretical and practical details of an algorithm which can be used to decide whether two given presentations for finite p-groups present isomorphic groups. The approach adopted is to construct a canonical presentation for each group. A description of the automorphism group of the p -group is also constructed.

The groups of order 128

Abstract A computer assisted determination of the 2328 groups of order 128 and of their 115 isoclinism families is described. Previous work by Rodemich on these groups is corrected. The provision of access to the group descriptions obtained is discussed.

Classifying 2-groups by coclass

Now that the conjectures of Leedham-Green and Newrnan have been proved, we probe deeper into the classification of p-groups using coclass. We determine the pro-2-groups of coclass at most 3 and use these to classify the 2-groups of coclass at most 3 into families. Using extensive computational evidence, we make some detailed conjectures about the structure of these families. We also conjecture that the 2-groups of arbitrary fixed coclass exhibit similar behaviour.

Enumerating p-Groups

We present a new algorithm which uses a cohomological approach to determine the groups of order p n , where p is a prime. We develop two methods to enumerate p -groups using the Cauchy-Frobenius Lemma. As an application we show that there are 10 494213 groups of order 2 9 . 1991 Mathematics subject classification (Amer. Math. Soc.) : primary 20D15.

Constructive membership in black-box groups

Abstract We present an algorithm to reduce the constructive membership problem for a black-box group G to three instances of the same problem for involution centralizers in G. If G is a finite simple group of Lie type in odd characteristic, then this reduction can be performed in (Monte Carlo) polynomial time.

Constructive recognition of PSL(2,q)

Existing black box and other algorithms for explicitly recognising groups of Lie type over GF(q) have asymptotic running times which are polynomial in q, whereas the input size involves only log q. This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises PSL(2, q) explicitly. The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising SL(2,q) in its natural representation in polynomial time, given a discrete logarithm oracle for GF(q). The algorithm presented here takes as input a generating set for a subgroup G of GL(d, F) that is isomorphic modulo scalars to PSL(2, q), where F is a finite field of the same characteristic as GF(q); it returns the natural representation of G modulo scalars. Since a faithful projective representation of PSL(2, q) in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in q rather than in log q, elementary algorithms will recognise PSL(2, q) explicitly in polynomial time in these cases. Given a discrete logarithm oracle for GF(q), our algorithm thus provides the required polynomial time oracle for recognising PSL(2, q) explicitly in the remaining case, namely for representations in the natural characteristic. This leads to a partial solution of a question posed by Babai and Shalev: if G is a matrix group in characteristic p, determine in polynomial time whether or not Op(G) is trivial.