information:filtering
Conjugating permutations

In general, composing permutations written in cycle notation follows no easily described pattern – the cycles of the composition can be different from those being composed. However the cycle type is preserved in the special case of conjugating a permutation σ {\displaystyle \sigma } by another permutation π {\displaystyle \pi }, which means forming the product π σ π − 1 {\displaystyle \pi \sigma \pi ^{-1}}. Here, π σ π − 1 {\displaystyle \pi \sigma \pi ^{-1}} is the conjugate of σ {\displaystyle \sigma } by π {\displaystyle \pi } and its cycle notation can be obtained by taking the cycle notation for σ {\displaystyle \sigma } and applying π {\displaystyle \pi } to all the entries in it. It follows that two permutations are conjugate exactly when they have the same cycle type.