Group Presentation of the Direct Product.

Solution 1:

I am trying to give a proof of $1.2.6$ using free product of groups

Let $G_1\cong\frac{F_1}{R_1}$ and $G_2\cong\frac{F_2}{R_2}$ be presentations of two groups, where $\mid F_i\mid=a_i$(may be finite or infinite) with isomorphism $\phi_n,~n=1,2$. Then we claim that the the following sequence is exact:

$$1 \longrightarrow \langle R_1,R_2, [F_1,F_2]\rangle \longrightarrow F_1*F_2 \longrightarrow G_1\times G_2\longrightarrow 1$$

The universal property of free product of groups gives the map $\phi: F_1*F_2\to G_1\times G_2$ defined as $\phi \mid _{F_n}=i_n\phi_n$, where $i_n:G_n\to G_1\times G_2$ is usual inclusion map. Since the images of $\phi_n$ commutes in $G_1\times G_2$. Therefore $\langle R_1,R_2,[F_1,F_2]\rangle \subseteq Ker(\phi)$ similarly since $\phi$ factors through $\phi_1$ and $\phi_2$ so $Ker(\phi)\subseteq\langle R_1,R_2,[F_1,F_2]\rangle$ .