Combinatorial Properties of The Alternating & Dihedral Groups And Homomorphic Images of Fibonacci Groups

Abstract

Let X { n} n = 1,2,…, be a finite n -element set and let n n n S , A and D be the Symmetric, Alternating and Dihedral groups of n X , respectively. In this thesis we obtained and discussed formulae for the number of even and odd permutations (of an n − element set) having exactly k fixed points in the alternating group and the generating functions for the fixed points. Further, we give two different proofs of the number of even and odd permutations (of an n − element set) having exactly k fixed points in the dihedral group, one geometric and the other algebraic. In the algebraic proof, we further obtain the formulae for determining the fixed points. We finally proved the three families; F(2r,4r + 2), F(4r +3,8r + 8) and F(4r +5,8r +12) of the Fibonacci groups F(m , n) to be infinite by defining Morphism between Dihedral groups and the Fibonacci groups.