Show that \(A_n\) is generated by \(3\)-cycles for \(n \ge 3\).
Solution
Every element of \(A_n\) is a product of an even number of transpositions, hence a product of pairs of transpositions.
Each pair is a product of \(3\)-cycles. If the transpositions share a point, then \[ (a\ b)(b\ c)=(a\ b\ c). \] If they are disjoint, then \[ (a\ b)(c\ d)=(a\ b\ c)(b\ c\ d). \] If they are equal, their product is the identity.
Thus every element of \(A_n\) is a product of \(3\)-cycles. Conversely, every \(3\)-cycle is even because \[ (a\ b\ c)=(a\ c)(a\ b). \] Therefore the \(3\)-cycles generate \(A_n\) for \(n\ge3\).