Discussion
The permutation (12) (13) in \(S_{4}\) is similar to :
Answer: A
The permutation is similar (conjugate) to (2 3 4).
Compose right to left: (13) sends 1 to 3, and (12) leaves 3 alone, so 1 goes to 3; 3 goes to 1 and then to 2; 2 is fixed by (13) and sent to 1 by (12). So
\[ (12)(13) = (1\;3\;2) \]
a 3-cycle. In \(S_{n}\) two permutations are conjugate exactly when they have the same cycle type, so (12)(13) is conjugate to every 3-cycle, including (2 3 4). (Composing left to right gives (1 2 3), also a 3-cycle.)
(2 3) is a transposition, (1 2 3 4) a 4-cycle and (1 2)(3 4) a product of two disjoint transpositions, all different cycle types.
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