A group is termed a quasi-free group if it is a free product of cyclic groups of any order. Free groups being free product of infinite cyclic groups show that quasi-free groups are the free product of a free group and a number of finite cyclic groups. In this paper we show that for any group G there exists a group denoted H(G) and is called the associated group of G and show that if F is a quasi-free group and G is any group then H(F) is trivial and H(FG)  H(G). 2010 Mathematics Subject Classification: Primary 16U60, 20C05, 16S34, 20E06; Secondary 20C10, 20C40

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