Let $G$ be a group, $X$ a set of generators of G and assume that $f: X \rightarrow G$ is such that $G=\left
Then by the universal property there exists exactly one homomorphism $f^*:F(X)\rightarrow G$ such that $f=f^*\circ \mu$ (clear $f^*$ is an epimorpism), where $\mu:X\rightarrow F(X)$, $\mu(x)=[x],$ for $x\in X$.
If a set of relations $\mathcal{R}\subset \mbox{Ker}f^*$ is such that $Ker f^*$ is the smallest normal group in $F(X)$ containing $\mathcal{R}$ then the couple $(X, \mathcal{R})$ is called a genetic code of the group $G$ (of course it is not unique) and we write $G=\mbox{gr}(X\|\mathcal{R})$.
We have a very useful theorem:
Let $G=\mbox{gr}(X\|\mathcal{R})$ and $f: X \rightarrow G$ be a function satisfied conditions above. Assume that $K$ is a group and $h: f(X) \rightarrow K$ is an arbitrary mapping. Then the following conditions are equivalent:
(i) there exists exactly one homomorphism h':G\rightarrow K such that h'|_{f(X)}=h,
(ii) for any $[{x_1}^{\epsilon^1}\cdot...\cdot{x_n}^{\epsilon^n}]\in\mathcal{R}$, where $x_i\in X, \epsilon_i\in\{-1,1\}$ we have $(h\circ f)(x_1)^{\epsilon_1}\cdot...\cdot (h\circ f)(x_n)^{\epsilon_n}=1_K$
By the theorem above it is easy to see for example that $\mathbb{Z}_2\times\mathbb{Z}_2=gr(\{a,b\}\|a^2,b^2,aba^{-1}b^{-1})$ or more intuitively $\mathbb{Z}_2\times\mathbb{Z}_2=gr({a,b}\|a^2,b^2,ab=ba$).
Could anyone help me to prove that a genetic code of the additive group of $\mathbb{Q}$ is as follows $\mathbb{Q}=\mbox{gr}(x_1,x_2,...\|x_1=x_2^2,x_2=x_3^3,x_3=x_4^4...)$? Can one use for this purpose the theorem above? I will be grateful for your help.