Computing in Direct Powers of Expanded Groups. A Discussion of the Subpower Intersection Problem

Masters Thesis from the year 2013 in the subject Mathematics - Algebra, grade: 1,0, University of Linz (Institut für Algebra), language: English, abstract: In this thesis, expanded groups and the Subpower Intersection Problem for direct powers of groups will be discussed and then implemented in GAP. The first chapter is an introduction to universal algebra and GAP. Algebraic structures, especially groups, expanded groups, direct powers and some conceptsare defined as belonging to algebraic structures. Then an introduction to the programming language GAP, which was developed for computations in algebra, is given and GAP is used for proving a theorem about polynomial functions on groups of order 8. In the second chapter the theory of expanded groups, especially how to find the universe of an expanded group given by generators is explored. After that a data structure for computing with expanded groups in GAP and use this data structure for proving that not every normal subgroup of an expanded group is also an ideal and for counting the binary polynomials of a concrete expanded group is develeoped. The last chapter is about the Subpower Intersection Problem for direct powers of groups. First, the concept of strong generators for direct powers of groups is dintroduced and then it is discused how to find strong generators of the intersection of direct powers of groups. Finally, the algorithms for solving the problem in GAP and discuss compatible functions asan application of the Subpower Intersection Problem are implemented.