Exploring the Associated Groups of Quasi-Free Groups

Let G is a cyclic group. Then H(G) is a trivial group and if G = G₁∗ . . . ∗ G_n is the free product of the groups G₁, . . ., G_n, then [Formula presented]. Furthermore, if the groups G₁, G₂, . . ., G_nare 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 H(F)is trivial and [Formula presented], where a group is termed a quasi-free group if it is a free product of cyclic groups of any order.