Show that \(A_n\) is generated by \(3\)-cycles.
Solution
Every element of \(A_n\) is a product of an even number of transpositions. Pair those transpositions.
1. A product of two transpositions that share a point is a \(3\)-cycle (or the identity).
For distinct \(a,b,c\), \[ (a\ b)(b\ c)=(a\ b\ c), \] up to the convention for composition order. If the two transpositions are equal, their product is the identity.
2. A product of two disjoint transpositions is a product of two \(3\)-cycles.
For distinct \(a,b,c,d\), \[ (a\ b)(c\ d)=(a\ c\ b)(a\ c\ d). \] A direct check on the four moved points verifies the equality.
3. Therefore every even permutation is a product of \(3\)-cycles.
Write an element of \(A_n\) as a product of an even number of transpositions and group them into pairs. Apply 1 or 2 to each pair. Hence the \(3\)-cycles generate \(A_n\).