Fgmt Hans Cuypers

OpenMath Technology for Interactive Mathematical Documents

New technologies such as XML, XSL and both MathML and OpenMath make it possible to bring mathematics to the Internet. Indeed, OpenMath, a markup language for mathematical content, and OmDoc, its extension to mathematical documents, open a way of communicating mathematics between computers, between software applications and over the Internet without losing information. In this paper we describe the latest applications of OpenMath related technologies for Interactive Mathematical Documents. As an example we describe the way we incorporate these new technologies in a new version of Algebra Interactive, an interactive course on first and second year university algebra.

Linear groups generated by reflection tori

A reflection is an invertible linear transformation of a vector space fixing a given hyperplane, its axis, vectorwise and a given complement to this hyperplane, its center, setwise. A reflection torus is a one- dimensional group generated by all reflections with fixed axis and center. In this paper we classify subgroups of general linear groups (in arbitrary dimension and defined over arbitrary fields) generated by reflection tori.

Working with Finite Groups

Two common ways to describe groups are to present them by generators and relations or as automorphism groups of algebraic, geometric or combinatorial structures. (Think of linear groups acting on vector spaces, symmetry groups of regular polytopes, Galois groups etc.) An automorphism group of such a structure may also be considered to be a subgroup of the group of all permutations of the elements of that structure. Automorphism groups can thus be seen as permutation groups. Permutation groups are groups consisting of permutations of a set with composition of permutations as group multiplication. So, for example, we may view linear groups as permutation groups on the set of vectors of the underlying vector space (but this may not be the most efficient approach). The Todd-Coxeter coset enumeration method provides, among other things, a link between groups given by generators and relations on the one hand and permutation groups on the other.

The Small Mathieu Groups

In this project we use the tools and techniques from Chapter 8 to construct the small Mathieu groups M 10, M 11 and M 12. These groups were discovered by the French mathematician Emile Mathieu (1835–1890), who also discovered the large Mathieu groups M 22, M 23 and M 24. See [9, 10, 11]. They are remarkable groups: for example, apart from the symmetric and alternating groups, M 12 and M 24 are the only 5-transitive permutation groups. The group Mio has a normal subgroup of index 2 isomorphic to A6. The other five groups are among the 26 sporadic simple groups, occurring in the classification of finite simple groups. After Mathieu’s discovery of these five sporadic simple groups it took almost a century before the sixth sporadic simple group was found.

A characterization of the symplectic and unitary 3-transposition groups

Let D be a normal set of 3-transpositions in a group G. For each d ∈ D, let Dd be the set of elements e in D such that the order of de equals 2. Then we can define an equivalence relation ⊺ on D by dre if and only if Dd ⋃ {d} = De ⋃ {e}. We characterize the symplectic and unitary groups over GF(2), respectively, GF(4) generated by their transvections as groups generated by a class D of 3-transposition with r trivial on D, but not when restricted to Dd for some d ∈ D.

Explorations with the Icosahedral Group

In Project 6 we have encountered a way to construct groups via a permutation representation. In the early seventies this has been one of the main tools in constructing sporadic simple groups. However, the permutation representations of the large sporadic simple groups like the so-called Monster and Baby-Monster have too high degree to put them on a computer, see the Atlas [1]. For these groups one has to use different methods. Many of these large sporadic simple groups, including the Monster (see [4]), have been constructed as a matrix group. In this project we will show by means of a small example how one may proceed to construct a group as a matrix group.