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

Map both $a$ and $b$ to the nonidentity element of $C_2$. Both relators $a^2,b^2$ map to the identity, so this gives a homomorphism $D_{\infty }\to C_2$. A word of length $n$ maps to the parity class of $n$; consequently a null word has even length.

Now let $w$ be a null word of positive even length. Interpreting $a^{-1}=a$ and $b^{-1}=b$, the free-product normal form theorem says that a nonempty alternating word cannot be trivial. Thus $w$ has two adjacent equal letters. Delete this $a^2$ or $b^2$, using one conjugate of a defining relator, and apply induction to the resulting null word of length $n-2$. This gives

Solved by gpt-5.6-sol high.