# What do all the $k$-cycles in $S_n$ generate?

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.

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.

• Nice proof. ---- Commented Nov 27, 2011 at 18:00

Yes, $$k$$ -cycles generate the symmetric group when $$k$$ is even and alternating group, when $$k$$ is odd. As you've said, for $$k=2$$ you know the answer. Suppose $$k>2$$.
$$(1\space 2\space\ldots,k)(k\space\ldots 3\space 1\space 2)=(1\space 3\space 2)$$
Similarly you can get any $$3$$-cycle.

Suppose $$a$$ is an even element of $$S_n$$. Then, as you know, $$a$$ is a product
$$(k_1\space k_2)(k_3\space k_4)\ldots(k_{4l-1}\space k_{4l})$$
of $$2l$$ transpositions. But product of $$2$$ transpositions can be written using $$3$$-cycles:
$$(k_1\space k_2)(k_2\space k_3)=(k_1\space k_2)(k_2\space k_3)\circ(k_2\space k_3)(k_3\space k_4).$$ Any of two products, separated by $$\circ$$ in right hand side, is either a $$3$$-cycle or unity.

If $$a$$ is odd and $$k$$ is even, than $$a$$ multiplied by any $$k$$-cycle is even, so we can apply the previous algorithm to it.

• Pardon my ignorance, but may I ask, was the assumption on parity of $k$ used only to show that $k$ cycles also generate odd permutations as well ? Commented Sep 27, 2023 at 19:32
• @Invisible The $k$-cycles with even $k$ are odd permutations: in this case they generate the whole $S_n$ including the odd permutaitons. The $k$-cycles with odd $k$ are even permutations, so they only generate the subgroup of all even permutations in $S_n$. This subgroup is also known under the name "alternating group". So, indeed, the difference between even and odd case is whether $k$-cycles generate odd permutations (they always generate all even permutations for $k\geq 2$). Commented Sep 28, 2023 at 1:15