Final Program

We study the problem of decomposing a finite dimensional module over an algebra. An algorithm is given to compute a decomposition, which has been implemented in the C-Meat-Axe. Finally, we compare our implementation with the MAGMA-procedure IndecomposableSummands.

We analyze the structure of the groups of cube-free order and, based on this, we present an algorithm for constructing these groups up to isomorphism.

???

Residue class-wise affine groups are a type of permutation groups over infinite sets which are accessible to computations. They naturally arise as groups of homoeomorphisms of suitable rings endowed with a topology by taking the set of all residue classes as a basis. The talk will provide some information about the structure of the group of all residue class-wise affine permutations of the integers without going into technical details, and illustrate this with some nice examples of elements of the mentioned group.

For del Pezzo surfaces of degree 9 the algebra associated to the group of automorphisms of the surface can be very effectively used for finding a rational parametrization. In the talk we explain how the construction of a simple module of the algebra gives an isomorphism of the surface and the projective plane.

We show how a problem about a certain moduli space of curves can be solved using modular invariant theory of finite groups and methods from computational commutative algebra and computational representation theory.

We prove a (weak) analogue of Broué's conjecture, exhibiting generalized perfect isometries in three families of groups where there is none in the sense of Broué (we study Suzuki Groups, Ree Groups of type G₂, and special unitary groups SU(3,q²) ).

I will describe an algorithm for constructing a multiplication table of a finitely-presented Lie ring that happens to be finite-dimensional.

Standard generators for the sporadic simple groups have been defined by R. A. Wilson for the Web Atlas of Finite Group Representations. We describe methods of checking that a pair of elements in a black box group (isomorphic to a sporadic simple group or the automorphism group of a sporadic simple group) is a pair of standard generators.

A subgroup covering of a group is a set of subgroups whose union contains every element of the group. Some people are interested in calculating the minimal sizes of subgroup coverings of simple groups.

In a project by Gramlich, Hoffman, Nickel and Shpectorov the simple connectedness of certain flag-transitive sub-geometries of spherical twin buildings is investigated. For the geometries belonging to the symplectic groups, the cases Sp(6,3), Sp(6,4), Sp(6,5), Sp(6,7) and Sp(8,2) are still open. A computer-based construction for the groups Sp(6,3), Sp(6,4), Sp(6,5) and Sp(8,2) is presented which yields the simple connectedness for the corresponding geometries using coset enumeration.