The circular division of the factor group when r is an odd number.

The benefit of this paper is to find the circular divison of the factor group when r is an odd number, where Q2r is denoted to Quaternion group of order 4r, such that for each positive integer r, there are two generators a and b for Q2r satisfies Q2r = {ah bk, 0 ≤ h ≤ 2r − 1, k = 0, 1} which has the following properties {a2r = b4 = I, barb−1 = a−r}. [ABSTRACT FROM AUTHOR]