On Artin Cokernel of the Quaternion Group Q_{2m} when m=2^h \cdot p_{1}^{r_1} \cdot p_{2}^{r_2} \cdots p_{n}^{r_n} such that p_i are Primes, g.c.d.(p_i, p_j)=1 and p_i \neq 2 for all i = 1, 2, ..., n, h and r_i any Positive Integer Numbers

Abstract

In this article, we find the cyclic decomposition of the finite abelian factor group AC(G)=\bar{R}(G)/T(G), where G=Q_{2m} and m is an even number and Q_{2m} is the quaternion group of order 4m. (The group of all Z-valued generalized characters of G over the group of induced unit characters from all cyclic subgroups of G). We find that the cyclic decomposition AC(Q_{2m}) depends on the elementary divisor of m. We have found that if m= p_{1}^{r_1} \cdot p_{2}^{r_2} \cdots p_{n}^{r_n} \cdot 2^h, p_i are distinct primes, then: AC(Q_{2m})=\bigoplus_{i=1}^{(r_1+1)(r_2+1)\cdots(r_n+n)(h+2)-1}C_2. Moreover, we have also found the general form of Artin characters table Ar(Q_{2m}) when m is an even number.