# 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

• Sahar Jaafar Mahmood Department of Mathematics, College of Computer Science and Information Technology, University of Al_Qadisiyah, Iraq
• Nesir Rasool Mahmood Department of Mathematics, College of Education for Girls, University of Kufa, Iraq
• Dhirgam Allawy Hussein Directorate of Education in Al_Qadisiyah, Iraq
Keywords: Artin cokernel, quaternion group, cyclic decomposition, Artin characters

### 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 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  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.

Published
2020-04-30
How to Cite
Mahmood, S. J., Mahmood, N. R., & Hussein, D. A. (2020). 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. Earthline Journal of Mathematical Sciences, 4(1), 169-188. https://doi.org/10.34198/ejms.4120.169188
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Articles