# 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 where and *m* is an even number and is the quaternion group of order 4*m*.

(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 depends on the elementary divisor of *m*. We have found that if are distinct primes, then:

Moreover, we have also found the general form of Artin characters table when *m* is an even number.

*Earthline Journal of Mathematical Sciences*,

*4*(1), 169-188. https://doi.org/10.34198/ejms.4120.169188

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