Papers
Improved Upper Bounds on the Spreads of Some Large Sporadic Groups
also available on the arxiv
Let G be a group. We say that G has spread r if for any set of distinct nontrivial elements {x1, . . . , xr} ⊂ G there exists an element y ∈ G with the property that hxi, yi = G for every 1 ≤ i ≤ r. Few bounds on the spread of finite simple groups are known. In this paper we present improved upper bounds for the spread of many of the larger sporadic simple groups, in some cases improving on the best known upper bound by several orders of magnitude.
- 72 Views
-
Like
Symmetric Representation of the Elements of the Conway Group ·0
Joint with my PhD supervisor, RT Curtis. This is a preprint version of the published article: Journal of Symbolic Computation, 44 (2009) p.1044-1067.
In this paper we represent each element of the Conway group ·0 as a permutation on 24 letters from the Mathieu group M24, followed by a sign change on a codeword of the binary Golay code (multiplication by a diagonal matrix taking the value -1 on the positions of a codeword and 1 otherwise), followed by a word of length at most four in a highly symmetric generating set.We describe an algorithm for multiplying elements represented in this way, that we have implemented in Magma. We include a detailed description of Λ4, the sets of 24 mutually orthogonal 4-vectors in the Leech lattice Λ often referred to as frames of reference or crosses, as they are instrumental to our procedure. In particular we describe the 19 orbits of M24 on these crosses.
- 75 Views
-
Like (1)
A Note On Monomial Representations Linear Groups
a slightly old version of a paper recently accepted by `Communications in Algebra'
A matrix is said to be monomial if every row and column has only one non-zero entry. Let G be a group. A representation \rho: G \rightarrow GL_n(C) is said to be a monomial representation of G if there exists a basis with respect to which \rho(g) is a monomial matrixfor every g\in G. We use elementary methods to classify the irreducible monomial representations of the groups L_2(q), L_3(q) and their natural decorations.
- 69 Views
-
Like (1)
Recent Progress in the Symmetric Generation of Groups
A survey article written for the proceedings of the conference `Groups St Andrews 2009'
Many groups posses 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.
- 18 Views
-
Like
Symmetric Generation of Coxeter Groups
Joint with J\"{u}rgen M\"{u}ller of RWTH Aachen and accepted by Archiv der Mathematik.
We provide involutory symmetric generating sets of finitely generated Coxeter groups, fulfilling a suitable finiteness condition, which in particular is fulfilled in the finite, affine and compact hyperbolic cases.
Symmetric Presentations of Coxeter Groups
A slightly old version of a paper submitted to the "Proceedings of the Edinburgh Mathematical Society"
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.
- 53 Views
-
Like (1)
Some Design Theoretic Results on the Conway Group ·0
submitted to `The Electronic Journal of Combinatorics'
Let \Omega be a set of 24 points with the structure of the (5,8,24) Steiner system, S, defined on it. The automorphism group of S acts on famous Leech lattice, as does the binary Golay code defined by S. Let A,B \subset\Omega be subsets of size four (“tetrads”). The structure of S forces each tetrad to define a certain partition of into six tetrads called a sextet. For each tetrad Conway defined a certain automorphism of Leech lattice that extend the group generated by the above to the
automorphism group of the lattice. For the tetrad A he denoted this automorphism \zeta_A. It is well known that for \zeta_A and \zeta_B to commute is sufficient to have A and B belong to the same sextet. We extend this to a necessary and sufficient condition, namely \zeta_A and \zeta_B will commute if and only if A\cup B is contained in a block of S. We go on extend this result to similar conditions for other elements of the group and show how neatly these results restrict to certain subgroups.
- 37 Views
-
Like (1)
Extensions of Symmetric Generating Sets
in preparation
In this paper we review existing methods of extending symmetric generating sets, namely Transitive Extensions and Subset Extensions before introducing a new approach using wreath products. We proceed to give examples of this new construction, most notably for the unitary groups U_3(2^r).
- 4 Views
-
Like
On the Symmetric Generation of Finite Groups
My PhD thesis.
Various aspects of the symmetric generation finite groups are discussed.
- 30 Views
-
Like
The exact spread of M12 is 9
preprint 2009
Let G be a group. We say that G has spread r if for any set of distinct non-trivial elements {x1,...,xr}\subset G there exists an element y\in G with the property that <xi, y> = G for every 1 0<i<r+1. The group G has exact spread r if it has spread r but not r + 1. The case where G is a finite simple group is particularly interesting since it is known that in this case the spread is at least 2. The precise value of the exact spread of a simple group is known in very few cases. Here we determine the precise value of the exact spread in the smallest sporadic group for which this is still unknown, the Mathieu group M12.
Add Comment