Are cyclic permutations always even?

Question

I just discovered the permutation symbol $\epsilon_{ijk}$ and was wondering if the set of cyclic permutations and even permutations are the same.

What does $\epsilon_{ijk}$ mean? – lhf Sep 02 '15 at 11:51 — lhf, Sep 02 '15 at 11:51

lhf · Accepted Answer · 2015-09-02T12:10:39.013

1

No, the cycle $(1234)=(14)(13)(12)$ is odd.

More generally, cycles of even order are odd permutations and cycles of odd order are even permutations because $(12\cdots n)=(1n)\cdots(13)(12)$ is a product of $n-1$ transpositions.

edited Sep 02 '15 at 12:10

answered Sep 02 '15 at 11:50

lhf

216,483

An even simpler example is the cycle $(12)$, which is odd. – 5xum Sep 02 '15 at 11:53
Oh I guess I meant for three dimensions. – Malz Sep 02 '15 at 11:55
You might want to add the formula: $;\varepsilon(\gamma)=(-1)^{l(\gamma)-1}$, if $;l(\gamma)$ is is the length of a cycle $\gamma$. – Bernard Sep 02 '15 at 11:56
@Malz Well all cycles of length $3$ are even, sure. – 5xum Sep 02 '15 at 14:52

Are cyclic permutations always even?

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