Alexander Hulpke

On Transitive Permutation Groups

We assign names and new generators to the transitive groups of degree up to 15, reflecting their structure.

Constructing transitive permutation groups

This paper presents a new algorithm to classify all transitive subgroups of the symmetric group up to conjugacy. It has been used to determine the transitive groups of degree up to 30.

The number of Latin squares of order 11

Constructive and nonconstructive techniques are employed to enumerate Latin squares and related objects. It is established that there are (i) 2036029552582883134196099 main classes of Latin squares of order 11; (ii) 6108088657705958932053657 isomorphism classes of one-factorizations of K 11,11 ; (iii) 12216177315369229261482540 isotopy classes of Latin squares of order 11; (iv) 1478157455158044452849321016 isomorphism classes of loops of order 11; and (v) 19464657391668924966791023043937578299025 isomorphism classes of quasigroups of order 11. The enumeration is constructive for the 1151666641 main classes with an autoparatopy group of order at least 3.

Techniques for the Computation of Galois Groups

Galois theory stands at the cradle of modern algebra and interacts with many areas of mathematics. The problem of determining Galois groups therefore is of interest not only from the point of view of number theory (for example see the article [39] in this volume), but leads to many questions in other areas of mathematics. An example is its application in computer algebra when simplifying radical expressions [32].

Computing Subgroups Invariant Under a Set of Automorphisms

This article describes an algorithm for computing up to conjugacy all subgroups of a finite solvable group that are invariant under a set of automorphisms. It constructs the subgroups stepping down along a normal chain with elementary abelian factors.

Block systems of a Galois group

We describe an algorithm to compute subfields of an algebraic number field as block systems of its Galois group. It relies only on symbolic computations and avoids numerical approximations.

Conjugacy classes in finite permutation groups via homomorphic images

The lifting of results from factor groups to the full group is a standard technique for solvable groups. This paper shows how to utilize this approach in the case of non-solvable normal subgroups to compute the conjugacy classes of a finite group.

Computing normal subgroups

This note presents a new algorithm for the computation of the set of normal subgroups of a finite group. It is based on lifting via homomorphic images.

Zariski density and computing in arithmetic groups

For

EFFICIENT SIMPLE GROUPS

n > 2