TITLE: What do all the $k$-cycles in $S_n$ generate?
QUESTION [18 upvotes]: 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.

REPLY [34 votes]: If $n\geq5$, then the only normal subgroups of the symmetric group $S_n$ are the trivial group, the alternating group and the symmetric group itself. Since the $k$-cycles form a full conjugacy class, it follows that the subgroup they generate is normal. This determines everything if $n \geq 5$.
More specifically: the $k$-cycles in $S_n$ generate the alternating group if $k$ is odd and $k \ne 1$; they generate the full symmetric group if $k$ is even.