The Cyclic Decomposition of the Factor Group cf (Dn×C3 ,Z )/R(Dn×C3 ) When n is an Odd Number

Abstract

Let D_n be the dihedral group , C₃ be the cyclic group of order 3 and D_nh is the direct product group of D_n and C₃ (i.e. D_n×C₃) . Let  cf(D_n ×C₃,Z) be the abelian group of Z-valued class functions of the group D_n ×C₃ . The intersection cf(D_n ×C₃,Z) with the group of all generalized characters of D_n ×C₃ which is denoted by (D_n ×C₃) , is a normal subgroup of cf(D_n ×C₃,Z) denoted by

(D_n ×C₃) , then factor group cf(D_n ×C₃,Z)/(D_n ×C₃) is a finite abelian  group denoted by K(D_n ×C₃) .

The problem of determining the cyclic decomposition of the group

K(D_n ×C₃) seem to be untouched .

The  aim of this paper is to find the cyclic decomposition of this group.

We find that when n is an odd number such that  , where all  are distinct  primes , then

K(D_n ×C₃)=C₄C₃ C₈.