conjugacy classes in the symmetric group
We construct the cycle decomposition for . Let , and regard as acting on . Let be the subgroup of generated by . Then acts on , so by the orbit-stabilizer theorem, it partitions into a unique set of orbits. In addition, for any orbit , we have that
Now, is cyclic and thus is cyclic as well; its order is the smallest power of such that . Note also that . Using the explicit bijection above, we see that the unique cosets of in are
and that the elements of are
Thus on any orbit of size , is a -cycle. This shows that a cycle decomposition exists.
Uniqueness now follows easily, since the cycle determined by on an element of order is determined uniquely by construction from the element. Choosing a different element in the same orbit, say , gives instead
which is the same cycle permuted left by . ∎
We can thus write
The above theorem proves that the cycle type is well-defined.
Two permutations are conjugate if and only if they have the same cycle type.
Assume first that and are conjugate; say . Write as a product of disjoint cycles
To show that and have the same cycle type, it clearly suffices to show that if follows in the cycle decomposition of , then follows in the cycle decomposition of . But suppose . Then
and we are done.
Now suppose and have the same cycle type. Write the cycle decomposition for each permutation in such a way that the cycles are listed in nondecreasing order of their length (including cycles of length 1). We then have (for example)
Define to be the permutation that takes to . Clearly , since each of appears exactly once among the and once among the . But also, since the cycle types of and match, we see that
where , are the ”‘next”’ elements in their respective cycles. Thus and we are done. ∎
The number of conjugacy classes in is the number of partitions of .
Each distinct cycle type in represents a distinct partition of , and each cycle type represents a conjugacy class. The result follows. ∎
We can give an explicit for the size of each conjugacy class in .
Suppose , and let be the distinct integers (including if applicable) in the cycle type of , and let there be cycles of order in . (Thus ). Then the number of conjugates of is exactly
The size of a conjugacy class is the number of cycles of the given cycle type. Choose a cycle type, and order the cycles in some order. Consider the possible assignments of the integers from to into the ”‘holes”’ in the cycles. Call two such arrangements equivalent if they define the same permutation. It is clear that this is an equivalence relation, and that the relation partitions the arrangements. We will determine the size of each equivalence class.
Consider a particular arrangement (i.e. permutation), and consider the cycles of order in that permutation. Each of the cycles can be written in different ways (by starting with a different element). Also, the cycles themselves can appear in any of possible orders while still representing the same permutation. Thus if, for example, and the first cycle contains while the second contains , that is the same permutation as if the first cycle contained and the second contained . So there are equivalent permutations considering the cycles of order . So the total number of permutations, considering each of the possible cycle orders, equivalent to the given permutation is
and the result follows. ∎
For example, let and let’s compute the size of the conjugacy class containing the permutation . In this case, ,
and the skeleton for the permutation is
There are ways of placing into this skeleton. However, given a particular choice, say , the first -cycle may be written as or as , while the second may be written as either or . In addition, either pair can appear first (). The number of choices for ways to represent the -cycles is thus . The formula in the theorem indeed predicts that there are permutations with this cycle type; in fact, they are (omitting the trivial -cycles):
- 1 D.S. Dummitt, R.M. Foote, Abstract Algebra, Wiley, 2004.
|Title||conjugacy classes in the symmetric group|
|Date of creation||2013-03-22 17:16:24|
|Last modified on||2013-03-22 17:16:24|
|Last modified by||rm50 (10146)|