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AbstractWe 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 earlier algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.

ABSTRACTWe give a method to describe all congruence images of a finitely generated Zariski dense group . 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 nas long as Hcontains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.

We survey the legacy of L. G. Kovács in linear group theory, with a particular focus on classification questions.

We present a classification of the nilpotent primitive subgroups of $\mathrm{GL}(n,q)$, up to conjugacy in $\mathrm{GL}(n,q)$. Groups in the classification are specified explicitly by generating sets of matrices, where each generating set has size at most 2. Additionally we give an algorithm designed to provide electronic access to the classification for any $n$ and $q$.

This paper deals with the poset of submodules of the direct sum of two modules. A well-known theorem of Remak (1930) gives a description of the set of submodules of the direct sum in terms of isomorphisms between sections of the direct summands. Given two submodules described in these terms, it is difficult in general to determine whether one is contained in the other. In this paper, we show how to determine whether one is contained in the other as maximal submodule. That is, we give a method of describing the edge set of the Hasse diagram of this poset.