centralizer of a k-cycle

Theorem 1.

Let σ be a k-cycle in Sn. Then the centralizerMathworldPlanetmathPlanetmathPlanetmath of σ is


where Sn-k is the subgroupMathworldPlanetmathPlanetmath of Sn consisting of those permutationsMathworldPlanetmath that fix all elements appearing in σ.


This is fundamentally a counting argument. It is clear that σ commutes with each element in the set given, since σ commutes with powers of itself and also commutes with disjoint permutations. The size of the given set is k(n-k)!. However, the number of conjugates of σ is the index of CSn(σ) in Sn by the orbit-stabilizer theorem, so to determine |CSn(σ)| we need only count the number of conjugates of σ, i.e. the number of k-cycles.

In a k-cycle (a1ak), there are n choices for a1, n-1 choices for a2, and so on. So there are n(n-1)(n-k+1) choices for the elements of the cycle. But this counts each cycle k times, depending on which element appears as a1. So the number of k-cycles is




so that


and we see that the elements in the list above must exhaust CSn(σ). ∎

Title centralizer of a k-cycle
Canonical name CentralizerOfAKcycle
Date of creation 2013-03-22 17:18:00
Last modified on 2013-03-22 17:18:00
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 20M30