Browsing by Author "Supervisor : Mihoubi, D"

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    Group actions on a set and P𝐨́lya’s Enumeration Method
    (University of M'sila, 2016-06-10) KADI, Saad Eddine; Supervisor : Mihoubi, D
    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.