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At the very beginning of the cyclic permutation Wikipedia article. There is a brief example of a cyclic permutation vs a non-cyclic permutation.

The article says that:

$\lbrace1,2,3,4 \rbrace \rightarrow \lbrace 3,4,2,1 \rbrace$

is cyclic, whilst:

$\lbrace1,2,3,4 \rbrace \rightarrow \lbrace 3,4,1,2 \rbrace$

is not cyclic.

I would have thought it was the other way around! Applying the method of taking the last entry and sticking it on the beginning twice (as in the Wolfram Mathworld article), gives the second transformation above. However taking the last entry and sticking it on the front (or taking the first entry and sticking it on the end) does not give the first transformation above, no matter how often such a swap is used.

user15766
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    I think the first one is $1\to3,2\to4,3\to2,4\to1$, or $1\to3\to2\to4\to1$ is cyclic, while the second one is $1\to3\to1$ and $2\to4\to2$ – Empy2 Oct 02 '15 at 23:10
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    First of all, that notation is terrible. There are lots of standard ways to write permutations, but that one is not one of them. (And wikipedia does not use that notation.) – Thomas Andrews Oct 02 '15 at 23:17
  • Thanks Michael! I see how I misunderstood the concept now. Thomas, apologies for my notation, I'm not a mathematician. As you might have guessed. – user15766 Oct 02 '15 at 23:19

1 Answers1

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No, the example is correct.

There's a traditional notation for cycles: $(a,b,c,d)$ is the permutation that sends $a$ to $b$, $b$ to $c$, $c$ to $d$ and $d$ to $a$. You can verify that that first permutation is $(1,3,2,4)$. So it's a single cycle; a cyclic permutation.

On the other hand the second sends $1$ to $3$ and $3$ back to $1$. It sends $2$ to $4$ and $4$ back to $2$. So it is $(1,3)(2,4)$, not a single cycle.

David C. Ullrich
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