If n >= 3, show that every element in An is a product of 3-cycles.
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Pair the transpositions together. Each pair has one of each possible forms: $\displaystyle (12)(12),(12)(23),(12)(34)$. Notice that $\displaystyle (12)(12) = (123)(123)(123)$, $\displaystyle (12)(23) = (123)$, and $\displaystyle (12)(34) = (123)(234)$. Thus, altogether each permutation in $\displaystyle A_n$ is a product of $\displaystyle 3$-cycles.