Discussion
Consider the following statements :
(I) If \(n \ge 5\), then all cycles of length 3 are conjugates in An.
(II) Two elements of An that are conjugates in Sn are always conjugates in An also.
Which is correct ?
Answer: A
Only (I) is true.
(I) Any two 3-cycles are conjugate in \(S_{n}\) by some \(\sigma\). If \(\sigma\) is odd and \(n \ge 5\), there is a transposition \(\tau\) disjoint from the 3-cycle \(c\), and \(\sigma\tau\) is even and conjugates c to the same target, since \(\tau\) commutes with c. So all 3-cycles are conjugate in \(A_{n}\).
(II) is false: \(S_{n}\)-classes can split in \(A_{n}\). In \(A_{5}\), the 24 five-cycles form one class in \(S_{5}\) but split into two classes of 12 in \(A_{5}\).
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