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Say in $S_4$, I want to write $(1,2,3,4)$ as a product of 3 cycles? How should I do it? In general, how do you decompose larger cycles as the product of smaller cycles (but all having the same cycle length)?

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Well, your permutation is odd, and 3-cycles are even, so you cannot do it: any product of even permutations is even. – Henno Brandsma Mar 23 '11 at 10:24

You can't: 3-cycles belong to the subgroup $A_4$ which consists of the even permutations (that is, permutations that can be written as a product of an even number of transpositions), but $(1,2,3,4)$ is an odd permutation.

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The parity is a restricting factor, see the wikipedia entry. All 3-cycles are even (length is odd) and $(1,2,3,4)$ is odd, and any product of even permutations will be even... So better question: can we write any odd cycle as a product of 3-cycles? Trivially yes in $S_4$, but what about any $S_n$?

[edit] It turns out that $A_n$ is indeed generated by 3-cycles, which follows from $(a,b)(c,d) = (a, c, d)(a, b, d)$ and similar identities, so we can replace any pair of transpositions by a 3-cycle or 2 3-cycles. This gives a recipe for 3-cycles, that works for all even permutations.

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