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Abstract:
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The aim of this work is to investigate some algebraic and combinatorial aspects
of group factorizations. The main contribution of this dissertation is a set of new
results regarding factorization of groups, with emphasis on the nonabelian case. We
introduce a novel technique for factorization of groups, the so-called free mappings,
a powerful tool for factorization of a wide class of abelian and non-abelian groups.
By applying a certain group action on the blocks of a factorization, a number of
combinatorial and computational problems were noted and studied. In particular, we
analyze the case of the group Aut(Zn) acting on blocks of factorization of Zn. We
present new theoretical facts that reveal the numerical structure of the stabilizer of
a set in Zn, under the action of Aut(Zn). New algorithms for finding the stabilizer of
a set and checking whether two sets belong to the same orbit are proposed. |