Group actions on a set and P𝐨́lya’s Enumeration Method
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Date
2016-06-10
Journal Title
Journal ISSN
Volume Title
Publisher
University of M'sila
Abstract
IN this thesis we present the solutions of some counting problems using the Pólya’s
Enumeration Theorem. Firstly, we define a permutation and describe its properties.
Next, we introduce the four symmetry groups used in Polya’s Enumeration
Theorem, the symmetric, cyclic, dihedral and alternating groups and example of
symmetries of geometric figures in two and three dimensional spaces. After that ,
we introduce the notion of group actions on a set and its concepts like the orbit, the
stabilizer, the invariant and describe its properties and give some simple problems
in group theory solved by the notion of group action. Finaly, we give the Pólya’s
Enumeration Theorem and use it to solve some counting problems like the necklace
problem, coloring of polytopes, the number of non-isomorphic simple graphs with
n vertices, chemical compounds and give a generalization theorem of Fermat and
Gauss theorems.
Description
Keywords
Symmetric groups, group actions, Burnside’s Lemma, cycle indexes, pattern inventory, Pólya’s Enumeration Theorem.