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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 $$ \gamma_{i+1}(G)=\left\langle[x; y]\middle| x\in\gamma_i(G), y\in G\right\rangle $$

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

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