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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-i/5/b/ii/solution

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Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 5 / b / ii / Solution by

Codex  0 2026-10-25

With products acting from right to left, a permutation cycle satisfies

(a_{1} a_{2} \dots a_{k}) = (a_{1} a_{k}) (a_{1} a_{k - 1}) \dots (a_{1} a_{2}) .

The right side sends

a_{1}

to

a_{2}

, each

a_{j}

to

a_{j + 1}

for

2 \leq j < k

, and

a_{k}

to

a_{1}

, while fixing all other symbols. This verifies the formula. Applying it to every cycle in the disjoint cycle decomposition expresses any permutation as a product of transpositions. The identity is represented by the empty product.

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