Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 111 / 4 / a / Solution

Reversing a word for $w$ gives a word of the same length for $w^{-1}$ because every generator is an involution. Applying the same argument to $w^{-1}$ provesA shortest word for $w$ followed by $s_i$ gives ${\ell }_S(w{s}_i)\le {\ell }_S(w)+1$. Conversely, $w=(w{s}_i)s_i$ gives ${\ell }_S(w)\le {\ell }_S(w{s}_i)+1$. Hence