2015

An L2-quotient algorithm for finitely presented groups on arbitrarily
many generators

J. Algebra, 423 (2015), 1109–1142

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.

2014

Determining Aschbacher classes using characters

J. Aust. Math. Soc., accepted for publication

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.

2013

Fast recognition of alternating groups of unknown degree

J. Algebra, 392 (2013), 315–335

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.

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.

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Some word maps that are non-surjective on infinitely many finite
simple groups

J. Algebra, 392 (2013), 315–335

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.

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2012

Normal forms for matrices over uniserial rings of length two

J. Algebra, 358 (2012), 250–256

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.

2011

Computing minimal associated primes in polynomial rings over the
integers

J. Symbolic Comput., 46 (2011), no. 10, 1098–1104

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.

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2009

Generic extensions and generic polynomials for semidirect products

J. Algebra, 322 (2009), no. 11, 4040–4052

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.

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2008

The cohomology of the Heisenberg Lie algebras over fields of finite
characteristic

Proc. Amer. Math. Soc., 136 (2008), no. 11, 3803–3807

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.

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2014

Primary decomposition of zero-dimensional ideals over arbitrary fields

submitted

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(nd^{4}) 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.

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L3-quotients

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

Primary decomposition Z

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

See details

2012

An L3-U3-quotient algorithm for finitely presented groups

RWTH Aachen University (2012)

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.

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