Let $G_1$ and $G_2$ be two non abelian groups and $G_1 * G_2$ be the free product of these two groups. Can we define an epimorphism from $G_1 *G_2$ to the direct product $G_1 \times G_2$ of these groups.
I think for abelian groups it is easy to do, but for non how can we do this. Please help me regarding this.
On
Yes. (hint) the kernel of $\phi : G * H \rightarrow G \times H$ should be equal to subgroup of $ G * H$ generated by $\{g^{-1}h^{-1}gh \in G * H \ | \ g \in G \land h \in H\} = \{x\in G * H \ | \ \phi(x) = 0 \}$.
By definition of free product, whenever you have homomorphisms $f_i\colon G_i\to X$, then there is a unique homomorphism $G_1*G_2\to X$ that looks like $f_i$ on $G_i\subset G_1*G_2$. For $X=G_1\times G_2$, we can simple use the canonical inclusions $G_i\to G_1\times G_2$ as our $f_i$. Long story short, with $g_{i,j}\in G_i$, this maps $g_{1,1}g_{2,1}g_{1,2}g_{2,2}g_{1,3}\cdots g_{2,n}\mapsto \langle g_{1,1}g_{1,2}\cdots, g_{1,n}\,g_{2,1}g_{2,2}\cdots, g_{2,n}\rangle$.