Derivation Of The Cycle Index Formula Of The Affine ( ) Group As A Semidirect Product Of The Cyclic Groups An

Abstract

Abstract In this paper, we derive the cycle index formula of an affine( ) group acting on the elements of the field ℤ.The resulting cycle index is given as;( , )= || +( − 1) + ∑∅( ) |( ) !.We further express the resulting cycle in terms of the cycle index formula of"=#$ + % &ℎ()( % ∈ ℤ+ and the cycle index formula of " = #,$ &ℎ()( 0 ≠ , ∈ ℤ+ which the affine(p) group is a semi-direct product of. We also use the resulting cycle index formulas to solve some examples.