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In this paper, we present a uniform formula for the integration of polynomials over the unitary, orthogonal, and symplectic groups using Weingarten calculus. From this description, we further simplify the integration formulas and give several optimizations for various cases. We implemented the optimized Weingarten calculus in Maple in a package called IntHaar for all three compact groups. Here we will discuss its functions, provide instructions for the package, and produce some examples of computed integrals., Comment: 21 pages, 4 figures, software available at: http://www.math.rutgers.edu/~ag930/

We determine the minimal degree permutation representations of all finite groups with trivial soluble radical, and describe applications to structural computations in large finite matrix groups that use the output of the CompositionTree algorithm. We also describe how this output can be used to help find an effective base and strong generating set for such groups. We have implemented the resulting algorithms in Magma , and we report on their performance.

We describe an algorithm to compute a composition tree for a matrix group defined over a finite field, and show how to use the associated structure to carry out computations with such groups; these include finding composition and chief series, the soluble radical, and Sylow subgroups.

We provide a new, practical algorithm for deciding finiteness of matrix groups over function fields of zero characteristic. The algorithm has been implemented in GAP. Experimental results and extensions of the algorithm to any field of zero characteristic are discussed.

We describe practical algorithms for computing a polycyclic presentation and for facilitating a membership test for a polycyclic subgroup of GL(d,Q). A variation of this method can be used to check whether a finitely generated subgroup of GL(d,Q) is solvable or solvable-by-finite. We report on our implementations of the algorithms for determining a polycyclic presentation and checking solvability.

In this paper we describe a suite of new algorithms for studying polycyclic matrix groups—algorithms for testing membership and for uncovering the polycyclic structure of the group. We also describe an algorithm for deciding whether or not a group is solvable, which, in the important context of subgroups ofGL (n,Z), is equivalent to deciding whether or not a group is polycyclic. Algorithms were developed in Baumslag et al. (1991. The algorithmic theory of polycyclic-by-finite groups. J. Algebra, 142, 118–149) for all of these problems, but the algorithms in this paper represent a first step toward finding practical algorithms: experiments show that they are fast enough to be useful in studying some reasonably complex examples using current technology.

In 1962 Steinberg gave pairs of generators for all finite simple groups of Lie type. In this paper, for each finite orthogonal group we provide a pair of matrices which generate its derived group: the matrices correspond to Steinberg's generators modulo the centre. These generators have been implemented in the computer algebra system MAGMA and this completes the provision of pairs of generators in MAGMA for all (perfect) finite classical groups.

We consider the application of the Schreier-Sims algorithm and its variations to matrix groups defined over finite fields. We propose a new algorithm for the selection of the base points and demonstrate that it both improves the performance of the algorithm for a large range of examples and significantly extends the range of application. In particular, the random Schreier-Sims algorithm, with this enhancement, performs extremely well for almost simple groups.

We present a symbolic algorithm to solve for the zeros of a polynomial vector field equivariant with respect to a finite subgroup of O (n). We prove that the module of equivariant. polynomial maps for a finite matrix group is Cohen-Macaulay and give an algorithm to compute a fundamental basis. Equivariant normal forms are easily computed from this basis. We use this basis to transform the problem of finding the zeros of an equivariant map to the problem of finding zeros of a set of invariant polynomials. Solving for the values of fundamental polynomial invariants at the zeros effectively reduces each group orbit of solutions to a single point. Our emphasis is on a computationally effective algorithm and we present our techniques applied to two examples.

Let G be a finite group an F and arbitrary splitting field for G. Using the Amitsur-Levitzki theorem on the polynomial identities satisfied by a matrix ring over a commutative ring an algorithm is given for finding the minimal dimension t for which there is a simple t-dimensional FG-submodule U in a faithful FG-module M of dimension n

We develop methods for computing with matrix groups defined over a range of infinite domains, and apply those methods to the design of algorithms for nilpotent groups. In particular, we provide a practical nilpotency testing algorithm for matrix groups over an infinite field. We also provide algorithms to answer a number of structural questions for a nilpotent matrix group.The main algorithms have been implemented in GAP, for groups over the rational number field.