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AbstractThe benefit of this paper is to find the circular divison of the factor group when ris an odd number, where Q2ris denoted to Quaternion group of order 4r, such that for each positive integer r, there are two generators aand bfor Q2rsatisfies Q2r= {ahbk, 0 ≤ h≤ 2r− 1, k= 0, 1} which has the following properties {a2r= b4= I, barb−1= a−r}.