The Sym(n)-conjugacy class of an element of Alt(n) is an Alt(n)-conjugacy class if and only some representative commutes with an odd permutation

Let \sigma \in A_n. Show that all elements in the S_n-conjugacy class of \sigma are conjugate in A_n if and only if \sigma commutes with an odd permutation.


We wish to show that S_n \cdot \sigma = A_n \cdot \sigma if and only if some \tau \in C_{S_n}(\sigma) is an odd permutation.

In this previous exercise, we saw that S_n \cdot \sigma is a union of [S_n : A_n C_{S_n}(\sigma)] distinct A_n-conjugacy classes; moreover, it is clear that A_n \cdot \sigma \subseteq S_n \cdot \sigma.

Note the following.

  • \sigma commutes with an odd permutation
  • if and only if C_{S_n}(\sigma) \not\leq A_n
  • if and only if A_n C_{S_n}(\sigma) = S_n
  • if and only if [S_n : A_n C_{S_n}(\sigma)] = 1
  • if and only if S_n \cdot \sigma is a union of just one conjugacy class
  • if and only if S_n \cdot \sigma = A_n \cdot \sigma.
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