Constructing normalisers in finite soluble groups

AbstractThis paper describes algorithms for constructing a Hall π-subgroup H of a finite soluble group G and the normaliser NG(H). If G has composition length n, then H and NG(H) can be constructed using O(n4 log |G|) and O(n5 log |G|) group multiplications, respectively. These algorithms may be used to construct other important subgroups such as Carter subgroups, system normalisers and relative system normalisers. Computer implementations of these algorithms can compute a Sylow 3-subgroup of a group with n = 84, and its normaliser in 47 seconds and 30 seconds, respectively. Constructing normalisers of arbitrary subgroups of a finite soluble group can be complicated. This is shown by an example where constructing a normaliser is equivalent to constructing a discrete logarithm in a finite field. However, there are no known polynomial algorithms for constructing discrete logarithms

Radicals of group algebras and permutation representations of symplectic groups

In part A we consider three separate problems concerned with the radical of the group algebra of a finite group over a field of characteristic p dividing the order of the group. In Section I we characterise group-theoretically those soluble groups for which the radical of the centre of the group algebra is an ideal of the group algebra. In Section II we find a canonical basis for the radical of the centre of the group algebra of a finite group. In Section III we give an algorithm for determining the radical of the group algebra of a p-soluble group. We evaluate the result for groups of p-Iength one and prove that the exponent of the radical in this case is the same as for a Sylow p-subgroup. We show by examples that no similar result holds in the general case. In part B we quote a conjecture of J. A. Green's on characters of Chevalley groups and prove Theorem A (i) If the conjecture holds then, excepting for each r at most a finite number of values of q, the group PSp(2r+1,q) has no multiply transitive permutation representations for r > 1. (ii) PSp (4,q) has no multiply transitive permutation representations for q > 2, regardless of the conjecture

Fundamental Theorem of Algebra: A Survey of History and Proofs

Higher Educatio

OREGAMI: Software Tools for Mapping Parallel Computations to Parallel Architectures

22 pagesThe mapping problem in message-passing parallel processors involves the assignment of tasks in a parallel computation to processors and the routing of inter-task messages along the links of the interconnection network. We have developed a unified set of software tools called OREGAMI for automatic and guided mapping of parallel computations to parallel architectures in order to achieve portability and maximal performance from parallel systems. Our tools include a description language which enables the programmer of parallel algorithms to specify information about the static and dynamic communication behavior of the computation to be mapped. This information is used by the mapping algorithms to assign tasks to processors and to route communication in the network topology. Two key features of our system are (a) the ability to take advantage of the regularity present in both the computation structure and the interconnection network and (b) the desire to balance the user's knowledge and intuition with the computational power of efficient combinatorial algorithms

The foundation period in the history of group theory

Thesis (M.A.)--University of Illinois, 1911.Typescript