COMBINATORIAL PROPERTIES OF THE ALTERNATING & DIHEDRAL GROUPS AND HOMOMORPHIC IMAGES OF FIBONACCI GROUPS

Abstract

be a finite n -element set and let

Sn , An

and Dn

be the

Symmetric, Alternating and Dihedral groups of

X n , 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.