274 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 generation of M₂₂
This study will prove the Mathieu group M₂₂ contains two symmetric generating sets with control grougp L₃ (2). The first generating set consists of order 3 elements while the second consists of involutions
Specular sets
We introduce the notion of specular sets which are subsets of groups called
here specular and which form a natural generalization of free groups. These
sets are an abstract generalization of the natural codings of linear
involutions. We prove several results concerning the subgroups generated by
return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352
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 “simply laced” 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
String rewriting for Double Coset Systems
In this paper we show how string rewriting methods can be applied to give a
new method of computing double cosets. Previous methods for double cosets were
enumerative and thus restricted to finite examples. Our rewriting methods do
not suffer this restriction and we present some examples of infinite double
coset systems which can now easily be solved using our approach. Even when both
enumerative and rewriting techniques are present, our rewriting methods will be
competitive because they i) do not require the preliminary calculation of
cosets; and ii) as with single coset problems, there are many examples for
which rewriting is more effective than enumeration.
Automata provide the means for identifying expressions for normal forms in
infinite situations and we show how they may be constructed in this setting.
Further, related results on logged string rewriting for monoid presentations
are exploited to show how witnesses for the computations can be provided and
how information about the subgroups and the relations between them can be
extracted. Finally, we discuss how the double coset problem is a special case
of the problem of computing induced actions of categories which demonstrates
that our rewriting methods are applicable to a much wider class of problems
than just the double coset problem.Comment: accepted for publication by the Journal of Symbolic Computatio
Monomial Modular Representations and Construction of the Held Group
AbstractMonomial representations of familiar finite groups over finite fields are used to construct (infinite) semi-direct products of free products of cyclic groups by groups of monomial automorphisms. Finite homomorphic images of theseprogenitorsin which the actions on the group of automorphisms and on the cyclic components are faithful are sought. The smallest non-trivial images of this type are often sporadic simple groups. The technique is demonstrated by three examples over the fieldsZ3,Z5, andZ7, which produce the Mathieu groupM11, the unitary groupU3(5):2, and the Held group, respectively
Ordered Bell numbers, Hermite polynomials, Skew Young Tableaux, and Borel orbits
We give three interpretations of the number of orbits of the Borel
subgroup of upper triangular matrices on the variety \ms{X} of complete
quadrics. First, we show that is equal to the number of standard Young
tableaux on skew-diagrams. Then, we relate to certain values of a modified
Hermite polynomial. Third, we relate to a certain cell decomposition on
\ms{X} previously studied by De Concini, Springer, and Strickland. Using
these, we give asymptotic estimates for as the dimension of the quadrics
increases.Comment: We revised the manuscrip
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