On ArtinCokernel of The Group D_n×C_5 When n is an Odd Number

Abstract

The group of all Z-valuedgeneralized characters of G over the group of induced unit characters from all cyclic subgroups of G, AC(G)= (G)/T(G) forms a finite abelian  group, called ArtinCokernel of G .The problem of

finding the cyclic decomposition of Artincokernel  AC( ) has been considered in this paper when n is an odd number , we find that if n = p .p ..p , where  p₁,p₂,…, p_m are distinct primes and not equal to 2 , then :

AC( )=  C₂

= AC(D_n)  C₂

And we give the general form of Artin's characters table Ar( )when n is

an odd number .