arXiv subjects
Publications and source records attributed to Rebecca Waldecker.
Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we prove general structure results about finite groups that have a faithful transitive action with fixity 4. We also explain examples for many different possibilities of such actions.
This article revisits earlier work by the last author together with Kay Magaard. We correct mistakes in results about the Sylow 2- and 3-structure of groups that act with fixity 3 and one result about possible components, we improve some results and we add details.
This article revisits earlier work by the second author together with Kay Magaard. We correct several little results and we briefly discuss why, fortunately, the errors hardly affect our main theorems and in particular do not affect the classification of simple groups that act with fixity 2. As an addition to the submitted article, this version also contains GAP code in a little appendix at the end.
In this article we prove results about finite soluble groups that act with fixity 2 or 3.
In previous work, all finite simple groups that act with fixity 4 have been classified. In this article we investigate which ones of these groups act faithfully on a compact Riemann surface of genus at least 2 with fixity four in total and in such a way that fixity 4 is exhibited on at least one orbit. This is an extended version of the submitted article, including our GAP code.
Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we classify all finite simple groups that have a faithful transitive action with fixity 4, along with details about all possible such actions.
We describe a new algorithm for finding a canonical image of an object under the action of a finite permutation group. This algorithm builds on previous work using Graph Backtracking, which extends Jeffrey Leon's Partition Backtrack framework. Our methods generalise both Nauty and Steve Linton's Minimal image algorithm.
The year 1954 marks the beginning of a tradition of workshops at the MFO in the area of finite group theory. This is highlighted and put into context in this article (contribution to a book).
Backtrack search is a fundamental technique for computing with finite permutation groups, which has been formulated in terms of points, ordered partitions, and graphs. We provide a framework for discussing the most common forms of backtrack search in a generic way. We introduce the concept of perfect refiners to better understand and compare the pruning power available in these different settings. We also present a new formulation of backtrack search, which allows the use of graphs with additional vertices, and which is implemented in the software package Vole. For each setting, we classify the groups and cosets for which there exist perfect refiners. Moreover, we describe perfect refiners for many naturally-occurring examples of stabilisers and transporter sets, including applications to normaliser and subgroup conjugacy problems for 2-closed groups.
Motivated by the theory of Riemann surfaces, we classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed points on each non-regular orbit and at most four fixed points in total. In each case we also give information about the branching datum of the surface. There is a shorter version of this article (submitted for publication), and in this extended version we give many more details about the GAP code that we use for the calculations. We also explicitly include a lemma that we only quote in the short version, so we can explain how exactly the GAP calculations and the lemma work together.
We introduce a new framework for solving an important class of computational problems involving finite permutation groups, which includes calculating set stabilisers, intersections of subgroups, and isomorphisms of combinatorial structures. Our techniques generalise 'partition backtrack', which is the current state-of-the-art algorithm introduced by Jeffrey Leon in 1991, and which has inspired our work. Our backtrack search algorithms are organised around vertex- and arc-labelled directed graphs, which allow us to represent many problems more richly than do ordered partitions. We present the theory underpinning our framework, and we include the results of experiments showing that our techniques often result in smaller search spaces than does partition backtrack. An implementation of our algorithms is available as free software in the GraphBacktracking package for GAP.
We introduce a new framework for solving an important class of computational problems involving finite permutation groups, which includes calculating set stabilisers, intersections of subgroups, and isomorphisms of combinatorial structures. Our techniques are inspired by and generalise 'partition backtrack', which is the current state-of-the-art algorithm introduced by Jeffrey Leon in 1991. But, instead of ordered partitions, we use labelled directed graphs to organise our backtrack search algorithms, which allows for a richer representation of many problems while often resulting in smaller search spaces. In this article we present the theory underpinning our framework, we describe our algorithms, and we show the results of some experiments. An implementation of our algorithms is available as free software in the GraphBacktracking package for GAP.
We describe how orbital graphs can be used to improve the practical performance of many algorithms for permutation groups, including intersection and stabilizer problems. First we explain how orbital graphs can be integrated in partition backtracking, the current state of the art algorithm for many permutation group problems. We then show how our algorithms perform in practice, demonstrating improvements of several orders of magnitude for some problems.
We describe a family of new algorithms for finding the canonical image of a set of points under the action of a permutation group. This family of algorithms makes use of the orbit structure of the group, and a chain of subgroups of the group, to efficiently reduce the amount of search which must be performed to find a canonical image. We present both a formal proof of correctness of our algorithms and experiments on different permutation groups, which compare our algorithms with the previous state of the art.
We introduce orbital graphs and discuss some of their basic properties. Then we focus on their usefulness for search algorithms for permutation groups, including finding the intersection of groups and the stabilizer of sets in a group.
In this paper we analyze the structure of transitive permutation groups that have trivial four point stabilizers, but some nontrivial three point stabilizer. In particular we give a complete, detailed classification when the group is simple or quasisimple. This paper is motivated by questions concerning the relationship between fixed points of automorphisms of Riemann surfaces and Weierstrass points and is a continuation of the authors' earlier work.
The main theorem in this article shows that a group of odd order which admits the alternating group of degree 5 with an element of order 5 acting fixed point freely is nilpotent of class at most two. For all odd primes r, other than 5, we give a class two r-group which admits the alternating group of degree 5 in such a way. This theorem corrects an earlier result which asserts that such class two groups do not exist. The result allows us to state a theorem giving precise information about groups in which the centralizer of every element of order 5 has centralizer a 5-group.