The cyclic decomposition of the factor group CF(Dnh,Z)‎ R(Dnh)‎ when n is anodd number

Abstract EN

For fixed positive integer n³3 ,let Dn be the dihedral group, Dnh= Dn ÏC2 and cf(Dnh,Z) be the abelian group of Z-valued class functions of the group Dnh .The intersection of cf(Dnh,Z) with the group of all generalized characters of Dnh , R(Dnh) is a normal subgroup of cf(Dnh,Z) denoted by R (Dnh),then cf(Dnh,Z)/ R (Dnh) is a finite abelian factor group which is denoted by K(Dnh).

I n this paper ,we determine the cyclic decomposition of the finite abelian factor group cf(Dnh,Z)/R(Dnh) when n is an odd number, we find that the cyclic decomposition of K(Dnh) depends on the elementary divisor of n ,if n= p 1 1 a .p 2 2 a ..p m m a where p1 ,p 2 ,…, p m are distinct primes and 1 a , 2 a ,.., m a are positive integers, then ; K(Dnh)= 2 1 = Å i K(Dn) ( )( ) ( )2 1 1 1 1 1+ × 2 + + - = Å m i a a L a C2 Å K(C4).

And we find the rational valued characters table of the group Dnh.