# Group of finite rank are closed under quotient

We say that a group has finite rank r if every finitely generated subgroup can be generated by r elements and $r$ is the least positive integer with this property.

How can one prove that the class of finite rank groups is closed under quotients, i.e Let $H\triangleleft$G where $G$ of finite rank,then $G/H$ is of finite rank group.

## 1 Answer

Let L be a finitely generated subgroup of $G/H$. Denote by $x_1,...,x_p$ generators of $L$. Let $p:G\rightarrow G/H$ the canonical projection. Consider $L'$ the subgroup of $G$ generated by $p^{-1}(x_1),...,p^{-1}(x_p)$. Since the rank of $G$ is $r$ and $L'$ is finitely generated, $L'$ has $r$ generators, $y_1,...,y_r$. Remark that $L'\subset p^{-1}(L)$, thus $p(L')\subset L$. Since for $i=1,...p, p^{-1}(x_i)\in L'$, $x_i\in p(L')$. We deduce that $L\subset p(L')$ since $x_i, i=1,...,p$ generate $L$ and henceforth, $L=p(L')$. This implies that $L$ is generated by $r$ elements and the rank of $G/L$ is inferior or equal to $r$.