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  text: 'The [-4hf] $A_n$ is generated by its $3$-[49f cycles].\nThat is, every element of $A_n$ can be made by multiplying together $3$-cycles only.\n\n# Proof\n\nThe product of two [4cn transpositions] is a product of $3$-cycles: \n\n- $(ij)(kl) = (ijk)(jkl)$\n- $(ij)(jk) = (ijk)$\n- $(ij)(ij) = e$.\n\nTherefore any permutation which is a product of evenly-many transpositions (that is, all of $A_n$) is a product of $3$-cycles, because we can group up successive pairs of transpositions.\n\nConversely, every $3$-cycle is in $A_n$ because $(ijk) = (ij)(jk)$.',
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