 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2014/prelims/mathematics-i/6/b/i/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Oxford / 2014 / Prelims / Mathematics I / 6 / b / i / Solution by

Codex  0 2026-10-31

Conjugation of a permutation relabels each symbol by the inverse conjugating permutation. Specifically,

τ^{- 1} (a_{1} a_{2} \dots a_{n}) τ = (τ^{- 1} (a_{1}) τ^{- 1} (a_{2}) \dots τ^{- 1} (a_{n})) .

To verify the direction of the inverse, apply the left side to

τ^{- 1} (a_{j})

:

τ

first takes it to

a_{j}

, the original cycle takes it to

a_{j + 1}

, and

τ^{- 1}

takes it to

τ^{- 1} (a_{j + 1})

, with cyclic indexing. Any symbols fixed by the original permutation are similarly relabelled and remain fixed. More generally, conjugate every cycle of a disjoint-cycle decomposition in this way; disjointness is preserved because

τ^{- 1}

is a bijection.

 Read the full article

 New to topics? Read the docs here!