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The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition to the previous results and hierarchy, we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems.

We develop practical techniques to compute with arithmetic groups H ≤ SL ( n , Q ) for n > 2 . Our approach relies on constructing a principal congruence subgroup in H. Problems solved include testing membership in H, analyzing the subnormal structure of H, and the orbit-stabilizer problem for H. Effective computation with subgroups of GL ( n , Z m ) is vital to this work. All algorithms have been implemented in GAP .

We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group H ≤ SL ( n , Z ) for n ≥ 2 . More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of SL ( n , Q ) for n > 2 .

We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our previous algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties., 2019;15

Motivated by the success of quotient algorithms, such as the well-known p-quotient or solvable quotient algorithms, in computing information about finite groups, we describe how to compute finite extensions H ˜ of a finite group H by a direct sum of isomorphic simple Z p H -modules such that H and H ˜ have the same number of generators. Similar to other quotient algorithms, our description will be via a suitable covering group of H. Defining this covering group requires a study of the representation module, as introduced by Gaschutz in 1954. Our investigation involves so-called Fox derivatives (coming from free differential calculus) and, as a by-product, we prove that these can be naturally described via a wreath product construction. An important application of our results is that they can be used to compute, for a given epimorphism G → H and simple Z p H -module V, the largest quotient of G that maps onto H with kernel isomorphic to a direct sum of copies of V. For this we also provide a description of how to compute second cohomology groups for the (not necessarily solvable) group H, assuming a confluent rewriting system for H. To represent the corresponding group extensions on a computer, we introduce a new hybrid format that combines this rewriting system with the polycyclic presentation of the module.

We establish a canonical correspondence between connected quandles and certain configurations in transitive groups, called quandle envelopes. This correspondence allows us to efficiently enumerate connected quandles of small orders, and present new proofs concerning connected quandles of order p and 2p. We also present a new characterization of connected quandles that are affine.

We survey group-theoretic algorithms for finding (some or all) subgroups of a finite group and discuss the implementation of these algorithms in the computer algebra system GAP.

We ask if there is any right isosceles triangle in the hyperbolic plane, constructible with ruler and compass, for which the ratio of the hypotenuse to the side is rational. Although the question remains open, we determine that there are no such triangles with ratio $$1+\frac{m}{n}$$ , with $$n\le 12$$ .

This article describes a setup that – given a composition tree – provides functionality for calculation in finite matrix groups using the Trivial-Fitting approach that has been used successfully for permutation groups. It treats the composition tree as a black-box object. It thus is applicable to other classes of groups for which a composition tree can be obtained. As an example, we consider an effective algorithm for determining conjugacy class representatives.

This article describes a practical approach for determining the lattice of subgroups U < V < G between given subgroups U and G, provided the total number of such subgroups is not too large. It builds on existing functionality for element conjugacy, double cosets and maximal subgroups.

We investigate the state of the art in the computational determination and enumeration of the groups of small order. This includes a survey of the available algorithms and a discussion of their recent improvements. We then show how these algorithms can be used to determine or enumerate the groups of order at most 20, 000 with few exceptions and we discuss the orders in this range which remain as challenging open problems.

A generalized tetrahedron group is defined to be a group admitting a presentation $\langle x,y,z\mid x^l=y^m=z^n=W_1^p(x,y)=W_2^q(y,z)=W_3^r(x,z)=1\rangle$ where l, m, n, p, q, r≥ 2, each Wi(a,b) is a cyclically reduced word involving both a and b. These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized triangle groups. In this paper, we build on previous work to show that the Tits alternative holds for a generalized tetrahedron group G whenever (p,q,r)≠ (2,2,2), that is, G contains a non-abelian free subgroup or is solvable-by-finite. The term Tits alternative comes from the respective property for finitely generated linear groups over a field (see [16]).

We prove that the simple group which has order 372000 is efficient by providing an efficient presentation for it. This leaves one simple group with order less than one million, which has order 979200, whose efficiency or otherwise remains to be determined.

We give a presentation of length O (log2|G|) for the groups G ≅ PSU3(q). This result has applications in recent algorithms to compute the structure of permutation groups and matrix groups.

A generalized tetrahedron groups is defined to be a group admitting a presentation where l,m,n,p,q,r >= 2, each W_i(a,b) is a cyclically reduced word involving both a and b. These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized triangle groups. In this paper, we build on previous work to give a complete classification of all finite generalized tetrahedron groups., Mathematical Preprints;2008-03

This volume is the proceedings of a conference on Finite Geometries, Groups, and Computation that took place on September 4-9, 2004, at Pingree Park, Colorado (a campus of Colorado State University). Not accidentally, the conference coincided with the 60th birthday of William Kantor, and the topics relate to his major research areas. Participants were encouraged to explore the deeper interplay between these fields. The survey papers by Kantor, O'Brien, and Penttila should serve to introduce both students and the broader mathematical community to these important topics and some of their connections while the volume as a whole gives an overview of current developments in these fields.