Exploring the Associated Groups of Quasi-Free Groups

If G is a cyclic group, then H (G) is a trivial group and if G= G1*...*Gn is the free product of the groups G1,..., Gn, then H(G)= H(G1*...*Gn) isomorphic of H (G1)*...*H(Gn). Furthermore, if the groups G1, G2,..., Gn are cyclic groups, then H(G) is a trivial group. In this paper we show that for every group G there exists a group denoted H (G) and is called the associated group of G satisfying some important properties that as application we show that if F is a quasi-free group and G is any group, then F(H) is trivial and H(F*G) isomorphic H(G), where a group is termed a quasi-free group if it is a free product of cyclic groups of any order.