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Computational Group Theory
This sixth workshop on Computational Group Theory proved that its main themes âfinitely presented groupsâ, â-groupsâ, âmatrix groupsâ and ârepresentations of groupsâ are lively and active fields of research. The talks also presented applications to number theory, invariant theory, topology and coding theory
Linear groups and computation
Funding: A. S. Detinko is supported by a Marie SkĆodowska-Curie Individual Fellowship grant (Horizon 2020, EU Framework Programme for Research and Innovation).We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in this class of groups are surveyed. We illustrate the solution of hard mathematical problems by computer experimentation. Possible avenues for further progress are discussed.PostprintPeer reviewe
Linear groups and computation
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for this class of groups are surveyed. We illustrate the solution of hard mathematical problems by computer experimentation. Possible avenues for further progress are discussed
Recent advances in algorithmic problems for semigroups
In this article we survey recent progress in the algorithmic theory of matrix
semigroups. The main objective in this area of study is to construct algorithms
that decide various properties of finitely generated subsemigroups of an
infinite group , often represented as a matrix group. Such problems might
not be decidable in general. In fact, they gave rise to some of the earliest
undecidability results in algorithmic theory. However, the situation changes
when the group satisfies additional constraints. In this survey, we give an
overview of the decidability and the complexity of several algorithmic problems
in the cases where is a low-dimensional matrix group, or a group with
additional structures such as commutativity, nilpotency and solvability.Comment: survey article for SIGLOG New
Algorithms for group isomorphism via group extensions and cohomology
The isomorphism problem for finite groups of order n (GpI) has long been
known to be solvable in time, but only recently were
polynomial-time algorithms designed for several interesting group classes.
Inspired by recent progress, we revisit the strategy for GpI via the extension
theory of groups.
The extension theory describes how a normal subgroup N is related to G/N via
G, and this naturally leads to a divide-and-conquer strategy that splits GpI
into two subproblems: one regarding group actions on other groups, and one
regarding group cohomology. When the normal subgroup N is abelian, this
strategy is well-known. Our first contribution is to extend this strategy to
handle the case when N is not necessarily abelian. This allows us to provide a
unified explanation of all recent polynomial-time algorithms for special group
classes.
Guided by this strategy, to make further progress on GpI, we consider
central-radical groups, proposed in Babai et al. (SODA 2011): the class of
groups such that G mod its center has no abelian normal subgroups. This class
is a natural extension of the group class considered by Babai et al. (ICALP
2012), namely those groups with no abelian normal subgroups. Following the
above strategy, we solve GpI in time for central-radical
groups, and in polynomial time for several prominent subclasses of
central-radical groups. We also solve GpI in time for
groups whose solvable normal subgroups are elementary abelian but not
necessarily central. As far as we are aware, this is the first time there have
been worst-case guarantees on a -time algorithm that tackles
both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation,
with some new result
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