I was reading the following research article entitled "Asymptotic growth of finite groups""Asymptotic growth of finite groups" by S.Sarah Black. Here Prof.Professor Black statedmakes the following statement at the bottom of page 406:
Suppose $\Gamma$ isIndeed, given a f.g. residually finite group with a$\Gamma$ and finite generating set $S$. Let $(G_i)$ be a, consider the family $\mathcal{G} = (G_i)$, consisting of all finite quotient groupsquotients of $\Gamma$ with corresponding, with respective generating setsets $S_i$$\mathcal{S}$, where, for each i, $S_i$$S _i$ is the epimorphic image of $S$ under the mapping $\Gamma \rightarrow G_i$$\Gamma \twoheadrightarrow G_i$. Then the growth ofit is easy to see that $\Gamma$$\gamma_S^{\Gamma}$ and the family $(G_i)$ is$\gamma_{\mathcal{S}}^{\mathcal{G}}$ represent the same function.
Notation. Following the notation of [1, Section 1], for $n \in \mathbb{N}$ we have $$\gamma_S^{\Gamma}(n) := \left\vert \{g \in \Gamma \,\right\vert\, l_S(g) \le n \}\vert, \,\, \gamma_{\mathcal{S}}^{\mathcal{G}}(n) := \max_i \gamma_{S_i}^{G_i}(n).$$
If someone helps me get a detailed proof of the above statement, then I will be grateful to him/her forever.
[1] S. Black, "Asymptotic growth of finite groups", 1997.
