77 research outputs found
Recent Progress in the Symmetric Generation of Groups
Many groups possess highly symmetric generating sets that are naturally
endowed with an underlying combinatorial structure. Such generating sets can
prove to be extremely useful both theoretically in providing new existence
proofs for groups and practically by providing succinct means of representing
group elements. We give a survey of results obtained in the study of these
symmetric generating sets. In keeping with earlier surveys on this matter, we
emphasize the sporadic simple groups. ADDENDUM: This is an updated version of a
survey article originally accepted for inclusion in the proceedings of the 2009
`Groups St Andrews' conference. Since the article was accepted the author has
become aware of other recent work in the subject that we incorporate to provide
an updated version here (the most notable addition being the contents of
Section 3.4.)Comment: 14 pages, 1 figure, an updated version of a survey article accepted
for the proceedings of the 2009 "Groups St Andrews" conference. v2 adds
McLaughlin reference and abelian groups reference
Symmetric Presentations of Coxeter Groups
We apply the techniques of symmetric generation to establish the standard
presentations of the finite simply laced irreducible finite Coxeter groups,
that is the Coxeter groups of types An, Dn and En, and show that these are
naturally arrived at purely through consideration of certain natural actions of
symmetric groups. We go on to use these techniques to provide explicit
representations of these groups.Comment: This is the predecessor of arXiv:0901.2660v1. To appear in the
Proceedings of the Edinburgh Mathematical Societ
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Symmetric representation of elements of sporadic groups
Uses the techniques of symmetric presentations to manipulate elements of large sporadic groups and to represent elements of these groups in much shorter forms than their corresponding permutation or matrix representation. Undertakes to develop a nested algorithm and a computer program to manipulate elements of large sporadic groups
On the symmetric generation of finite groups
In this thesis we discuss some uses and applications of the techniques in Symmetric generation. In Chapter 1 we introduce the notions of symmetric generation. In Chapter 2 we discuss symmetric presentations defined by symmetric generating sets that are preserved by a group acting on them transitively but imprimitively. In Chapter 3 our attention turns to Coxeter groups. We show how the Coxeter-Moser presentations traditionally associated with the families of finite Coxeter groups of types A, D and E (ie the āsimply lacedā Coxeter groups) may be interpreted as symmetric presentations and as such may be naturally arrived at by elementary means. In Chapter 4 we classify the irreducible monomial representations of the groups L(q) and use these to define symmetric generating sets of various groups
Towards a combinatorial algorithm for the enumeration of isotopy classes of symmetric cellular embeddings of graphs on hyperbolic surfaces
Based on the recent mathematical theory of isotopic tilings, we present the, to the best of our knowledge, first algorithm for the enumeration of isotopy classes of cellular embeddings of graphs invariant under a given symmetry group on hyperbolic surfaces. To achieve this, we substitute the isotopy classes with combinatorial objects and propose different techniques, guided by structural results on the mapping class group of an orbifold and notions from computational group theory that ensure that the algorithm is computationally tractable. Furthermore, we extend data structures of combinatorial tiling theory to isotopy classes that lead to an actual implementation of the algorithm for symmetry groups generated by rotations. \\From the enumerated combinatorial objects, we produce a range of simple graphs on hyperbolic surfaces represented as symmetric tilings in the hyperbolic plane, illustrating the enumeration with examples and experimentally demonstrating the feasibility of the approach. These tilings are finally projected onto a family of triply-periodic surfaces that are relevant for the natural sciences
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Symmetric generation of finite groups
Advantages of the double coset enumeration technique include its use to represent group elements in a convenient shorter form than their usual permutation representations and to find nice permutation representations for groups. In this thesis we construct, by hand, several groups, including Uā(3) : 2, Lā(13), PGLā(11), and PGLā(7), represent their elements in the short form (symmetric representation) and produce their permutation representations
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Symmetric representations of elements of finite groups
This thesis demonstrates an alternative, concise but informative, method for representing group elements, which will prove particularly useful for the sporadic groups. It explains the theory behind symmetric presentations, and describes the algorithm for working with elements represented in this manner
Probabilistic symmetry reduction
Model checking is a technique used for the formal verification of concurrent systems. A major hindrance to model checking is the so-called state space explosion problem where the number of states in a model grows exponentially as variables are added. This means even trivial systems can require millions of states to define and are often too large to feasibly verify. Fortunately, models often exhibit underlying replication which can be exploited to aid in verification. Exploiting this replication is known as symmetry reduction and has yielded considerable success in non probabilistic verification.
The main contribution of this thesis is to show how symmetry reduction techniques can be applied to explicit state probabilistic model checking. In probabilistic model checking the need for such techniques is particularly acute since it requires not only an exhaustive state-space exploration, but also a numerical solution phase to compute probabilities or other quantitative values.
The approach we take enables the automated detection of arbitrary data and component symmetries from a probabilistic specification. We define new techniques to exploit the identified symmetry and provide efficient generation of the quotient model. We prove the correctness of our approach, and demonstrate its viability by implementing a tool to apply symmetry reduction to an explicit state model checker
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