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Why don't $3$-cycles generate the symmetric group? was asked earlier today. The proof is essentially that $3$-cycles are even permutations, and products of even permutations are even.

So: do the $3$-cycles generate the alternating group? Similarly, do the $k$-cycles generate the alternating group when $k$ is odd?

And do the $k$-cycles generate the symmetric group when $k$ is even? I know that transpositions ($2$-cycles) generate the symmetric group.

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