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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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2 Answers 2

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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.

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  • $\begingroup$ Nice proof. ---- $\endgroup$
    – user5262
    Commented Nov 27, 2011 at 18:00
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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.

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  • $\begingroup$ 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 ? $\endgroup$
    – PinkyWay
    Commented Sep 27, 2023 at 19:32
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    $\begingroup$ @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 $k\geq 2$). $\endgroup$
    – Fiktor
    Commented Sep 28, 2023 at 1:15

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