Abstract:
:" The purpose of this paper is to find The Cyclic Decomposition of The Factor Group cf(Q2m×D4,Z)/ R ?(Q2m×D4) when m is a prime number, where Q2m is denoted to Quaternion group of order 4n, such that for each positive integer n, there are two generators x and y for Q2m satisfies Q2m={xh yk, 0?h?2m-1, k=0,1}which has the following properties{x2m=y4=I, yxmy-1=x-m} and D4 is the Dihedral group of order 8 is generate by a rotation r of order 4 and reflection s of order 2 then 8 elements of D4 can be written as: {I*,r,r2, r3,s,sr,sr2,sr3}.
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