centralizer of a k-cycle
This is fundamentally a counting argument. It is clear that commutes with each element in the set given, since commutes with powers of itself and also commutes with disjoint permutations. The size of the given set is . However, the number of conjugates of is the index of in by the orbit-stabilizer theorem, so to determine we need only count the number of conjugates of , i.e. the number of -cycles.
In a -cycle , there are choices for , choices for , and so on. So there are choices for the elements of the cycle. But this counts each cycle times, depending on which element appears as . So the number of -cycles is
and we see that the elements in the list above must exhaust . ∎
|Title||centralizer of a k-cycle|
|Date of creation||2013-03-22 17:18:00|
|Last modified on||2013-03-22 17:18:00|
|Last modified by||rm50 (10146)|