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

Table of contents
- Solution 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}

proves

ℓ_{S} (w^{- 1}) = ℓ_{S} (w) .

A shortest word for

w

followed by

s_{i}

gives

ℓ_{S} (w s_{i}) \leq ℓ_{S} (w) + 1

. Conversely,

w = (w s_{i}) s_{i}

gives

ℓ_{S} (w) \leq ℓ_{S} (w s_{i}) + 1

. Hence

ℓ_{S} (w s_{i}) \in {ℓ_{S} (w) - 1, ℓ_{S} (w), ℓ_{S} (w) + 1} .

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