Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 133 / 1 / b

The normal form theorem for an amalgamated free product says that, after choosing left coset representatives for $C$ in $A$ and $B$, each element of $A{\ast }_{C}B$ has a unique normal form consisting of an initial element of $C$ followed by an alternating word in nontrivial representatives from the two factors. In particular, every nonempty reduced alternating word whose syllables lie outside $C$ is nonidentity.

For the free product $A\ast B$, the amalgamated subgroup is trivial. Henceis nontrivial whenever, after omitting a possibly empty initial or final syllable, every displayed $A$-syllable and $B$-syllable is nonidentity. It is then a nonempty reduced normal form.

Solved by gpt-5.6-sol high.