Sebastian Jambor

We generalize the Plesken-Fabiańska L2-quotient algorithm for finitely presented groups on two or three generators to allow an arbitrary number of generators. The main difficulty lies in a constructive description of the invariant ring of GL(2,K) on m copies of SL(2,K) by simultaneous conjugation. By giving this description, we generalize and simplify some of the known results in invariant theory. An implementation of the algorithm is available in the computer algebra system Magma.

Let Δ:G→GL(n,K) be an absolutely irreducible representation of an arbitrary group G over an arbitrary field K; let χ:G→K:g↦tr(Δ(g)) be its character. In this paper, we assume knowledge of χ only, and study which properties of Δ can be inferred. We prove criteria to decide whether Δ preserves a form, is realizable over a subfield, or acts imprimitively on K^n×1. If K is finite, this allows us to decide whether the image of Δ belongs to certain Aschbacher classes.

We show that there are only finitely many primes p such that PSL(2, p) has a minimal generating set of size four.

We present a constructive recognition algorithm to decide whether a given black-box group is isomorphic to an alternating or a symmetric group without prior knowledge of the degree. This eliminates the major gap in known algorithms, as they require the degree as additional input.
Our methods are probabilistic and rely on results about proportions of elements with certain properties in alternating and symmetric groups. These results are of independent interest; for instance, we establish a lower bound for the proportion of involutions with small support.

We provide the first examples of words in the free group of rank 2 that are not proper powers and for which the corresponding word maps are non-surjective on an infinite family of finite non-abelian simple groups.

This paper develops methods to describe the conjugacy classes of GL(n,R) on R^n×n for a serial ring R of length two. The main result is a reduction to a computation in the matrix algebra over the residue class field of R, which in some cases can be done theoretically without any actual computation.

An algorithm is presented to compute the minimal associated primes of an ideal in a polynomial ring over the integers. It differs from the known algorithms insofar as it avoids to compute Gröbner bases over the integers until the very end, thereby eliminating one of the bottlenecks of those algorithms.

This paper presents a generalization of a theorem of Saltman on the existence of generic extensions with group A⋊G over an infinite field K, where A is abelian, using less restrictive requirements on A and G. The method is constructive, thereby allowing the explicit construction of generic polynomials for those groups, and it gives new bounds on the generic dimension. Generic polynomials for several small groups are constructed.

We give explicit formulas for the cohomology of the Heisenberg Lie algebras over fields of finite characteristic. We use this to show that in characteristic two, unlike all other cases, the Betti numbers are unimodal.

We present two algorithms to compute the primary decomposition of zero-dimensional ideals. The algorithms are deterministic and mainly use linear algebra techniques, based on the FGLM approach. This allows us for the first time to give a complete complexity analysis of a primary decomposition algorithm. We show that the primary decomposition can be computed in O(nd⁴) field operations and d factorizations of univariate polynomials over the ground field, where n is the number of generators of the polynomial ring and d is the residue class dimension of the ideal. This result and the corresponding algorithm are valid for all fields. A publicly available computer implementation shows that the algorithms also perform very well in practice.

This is an implementation of the L2-quotient algorithm for finitely presented groups on arbitrarily many generators as described here.

This is an implementation of the L3-U3-quotient algorithm for finitely presented groups on two generators that I developed for my PhD thesis.

This is an implementation of the black-box A_n-recognition algorithm as described here. The verification part of the algorithm was developed by Jonathan Conder.

This is an implementation of the algorithms to compute the primary decomposition of zero-dimensional ideals defined over arbitrary fields as described here.

This is an implementation of the algorithm to compute the minimal associated prime ideals of arbitrary ideals defined over the integers as described here.

This is an implementation of the Pfister-Sadiq-Steidel algorithm to compute the primary decomposition of ideals in polynomial rings over the integers.

The thesis describes the development of an L3-U3-quotient algorithm for finitely presented groups on two generators, which finds all factor groups of a finitely presented group which are isomorphic to one of the groups PSL(3, q), PSU(3, q), PGL(3, q), or PGU(3, q). Here q is an arbitrary prime power which is not part of the input of the algorithm, so the algorithm finds all possible choices of q by itself. The motivation for such an algorithm is to study and understand finitely presented groups. Results from the middle of the last century show that central questions concerning these groups are undecidable in general. The most famous is the word-problem: there cannot exist an algorithm which decides the equality of two elements of a finitely presented group. Nevertheless, algorithms have been developed which give structural information about finitely presented groups. One class of such algorithms are the quotient algorithms, which find all factor groups of a given finitely presented group that have a certain structure. Until a few years ago, all of those algorithms only worked for a finite set of factor groups or for soluble groups. In 2009, Plesken and Fabiańska developed the first algorithm which can compute all factor groups in an infinite class of non-soluble groups. It determines all factor groups of a finitely presented group G which are isomorphic to one of the groups PSL(2, q) or PGL(2,q), simultaneously for any prime power q. The present thesis is a continuation of those ideas. For the formulation of the algorithm, various results from representation theory and from commutative algebra are needed. These are stated and proved in this work. The character of a representation has been an important tool in ordinary representation theory, i.e., of representations of finite groups over fields of characteristic zero. The results in this thesis show that it is still an invaluable tool for representations of arbitrary groups over arbitrary fields. A method in commutative algebra is the determination of all minimal associated prime ideals of a given ideal. This dissertation presents an algorithm which improves on the runtime of existing algorithms for important examples. The results in representation theory and in commutative algebra are applied to the L3-U3-quotient algorithm, but they are of general interest as well. The L3-U3-quotient algorithm is implemented in the computer algebra system Magma. This implementation is applied to several examples of finitely presented groups. Furthermore, results of this thesis are used to prove generalizations of theorems of P. Hall and Lubotzky.