Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 120 / 1 / e / Solution

Write the input word over an alphabet as $w=a_1\cdots a_n$. Starting with $z_0=\gamma$, obtain $z_i$ from $z_{i-1}$ by linear-time multiplication in an automatic structure with $a_i$. Mathematical induction givesThe time complexity of step $i$ is at most a constant times $|\gamma |+(i-1)N+1$. Summing the costs givesThe automatic structure for a group, $N$ and $\gamma$ are fixed, so the required representative isFor the empty word, return $\gamma$ in constant time.