## Alt(n) is generated by the set of all 3-cycles

Prove that $A_n$ is generated by the set of all 3-cycles for $n \geq 3$.

Let $n \geq 3$ and let $T = \{ (a\ b\ c) \ |\ 1 \leq a,b,c \leq n \}$ be the set of 3-cycles in $A_n$.

Note that $A_n$ contains $T$, so that $\langle T \rangle \leq A_n$.

Recall that $A_n$ consists of all permutations which can be written as an even product of transpositions; more specifically, $A_n$ is generated by the set of all products of two distinct 2-cycles. (Here we use the fact that $n \geq 3$.) Each product $\sigma$ of two 2-cycles has one of two forms.

If $\sigma = (a\ b)(c\ d)$, note that $\sigma = (a\ c\ b)(a\ c\ d)$.

If $\sigma = (a\ b)(a\ c)$, note that $\sigma = (a\ c\ b)$.

Thus $A_n \leq \langle T \rangle$, hence $A_n = \langle T \rangle$.