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We enumerate the 15768 15768 perfect groups of order up to 2 ⋅ 10 6 2\cdot 10^6 , up to isomorphism, thus also completing the missing cases reported by Holt and Plesken [Perfect groups, The Clarendon Press, Oxford University Press, New York, 1989]. The work supplements the by now well-understood computer classifications of solvable groups, illustrating scope and feasibility of the enumeration process for nonsolvable groups.

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 .

Our work was supported by a Marie Skłodowska-Curie Individual Fellowship grant under Horizon 2020 (EU Framework Programme for Research and Innovation), and Simons Foundation Collaboration Grant 244502. We give a method to describe all congruence images of a finitely generated Zariski dense group H ≤ SL (n,ℤ). The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n;if n=2 then we compute all congruence images only modulo primes. We propose a separate method that works for all n as long as H contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged. Postprint

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 describe an effective algorithm to compute a set of representatives for the conjugacy classes of Hall subgroups of a finite permutation or matrix group. Our algorithm uses the general approach of the so-called ‘trivial Fitting model’.

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]).

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.

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.

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

We describe an algorithm to determine representatives of the conjugacy classes of subgroups of a Trivial-Fitting group, this case being the one prior algorithms reduce to. As a subtask we describe an algorithm for determining conjugacy classes of complements to an arbitrary normal subgroup if the factor group is solvable.

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

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., Comment: Minor revision; to appear in Mathematics of Computation

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.

We show how to compute efficiently a lexicographic ordering for subgroups and cosets of permutation groups and, more generally, of finite groups with a faithful permutation representation.