# Surjectivity of a bilinear map between groups

Let $$G$$ a generic group and $$\gamma_i(G)$$ its lower central series defined as $$\begin{cases} \gamma_1(G)=G\\ \gamma_{i+1}(G)=\left[\gamma_i(G); G\right] \end{cases}$$ for every $$i\in \mathbb N$$. Set $$G'=[G; G]$$ i need to prove that this map $$f_i:\left(a\gamma_{i+1}(G), gG'\right)\in\frac{\gamma_i(G)}{\gamma_{i+1}(G)}\times\frac{G}{G'}\rightarrow[a; g]\gamma_{i+2}(G)\in\frac{\gamma_{i+1}(G)}{\gamma_{i+2}(G)}$$ is surjective for every $$i\in\mathbb N$$.

I already know that $$f_i$$ is a well-defined bilinear map between groups, but I don't understand why it's surjective because $$\gamma_{i+1}(G)=\left\langle[x; y]\middle| x\in\gamma_i(G), y\in G\right\rangle$$

Ok I solved by myself, because I don't nerd that $$f_i$$ is surjective but ora extension over the tensor product.