For each generator $t\in T$, let $|t|_S$ be its length in the generating set $S$, and put
$$
C=\max_{t\in T}|t|_S.
$$
A shortest $T$-word for $h_1^{-1}h_2$ has $d_T(h_1,h_2)$ letters. Replacing each letter by an $S$-word of length at most $C$ gives
$$
d_S(h_1,h_2)=|h_1^{-1}h_2|_S
\leq C|h_1^{-1}h_2|_T
=C\,d_T(h_1,h_2).
$$
Thus inclusion of any finitely generated subgroup is <Lipschitz continuous> for the corresponding <word metric>[word metrics].

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 133 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-133/3/a/solution