Choose any finite generating set $T$ of the <finitely generated group> $H$. Starting from a finite generating set $S_0$ of $G$, set
$$
S=S_0\cup T.
$$
Every $T$-word in $H$ of length at most $r$ is also an $S$-word in $G$ of length at most $r$. The inclusion of the corresponding word-metric balls is injective, so the <growth function of a finitely generated group> satisfies
$$
\boxed{\beta_{H,T}(r)\leq\beta_{G,S}(r)\qquad(r\geq0).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 3 / b / Solution

A <finitely generated group> has polynomial growth when, for some constants $C,D$, its <growth function of a finitely generated group> satisfies
$$
\beta_{G,S}(r)\leq C(1+r)^D.
$$
This property is independent of the finite generating set $S$.

Polynomial growth of a group

{title2=$\beta_{G,S}$}

For a <finitely generated group> $G$ with finite generating set $S$, the growth function $\beta_{G,S}(r)$ is the number of group elements of <word metric> distance at most $r$ from the identity. Different finite generating sets change its scale only by multiplicative constants.

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