If $x \in X$, the set $x^G = \{ x^g \in X: g \in G\}$ is called the orbit of $x$ under the action of $G$.

I wanted to look up some example and stumbled upon this one:

For the permutation group $G = \{(1234),(2134),(1243),(2143)\}$ the orbit of $1$ and $2$ is $\{1,2\}$ and the orbit of $3$ and $4$ is $\{3,4\}$.

(Link: http://mathworld.wolfram.com/GroupOrbit.html).

By looking at the provided definition, I would have guessed that the orbit of $1$ for example is $\{2,3,4\}$. My reasoning: take $1 \in X$, and take the first permutation in $G$. This one maps $1$ t0 $2$. The second permutation in $G$ maps $1$ to $3$, and so on. Meaning that the set of possible images of $x$ under the action of $G$ equals $\{2,3,4\}$.

Why is my reasoning incorrect?


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    $\begingroup$ It's simple: $G$ is not a group. For instance $(1234)^2=(13)(24)$ is not in $G$. $\endgroup$ – Captain Lama Apr 29 at 16:47
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    $\begingroup$ I think you made a confusion with the double-transposition group. $\endgroup$ – Captain Lama Apr 29 at 16:48
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    $\begingroup$ I think they are using a linear ordering notation for the permutations rather than a cycle notation (i.e. they are giving the images of $1,2,3,4$ in that order). For example $(1234)$ is the identity, $(2134)$ is the transposition $(1,2)$, and $(2143)$ would be, in cycle notation $(1,2)(3,4)$. Then what they say about orbits is right, and moreover, in cycle notation, the four given permutations don't form a group. $\endgroup$ – Ned Apr 29 at 16:52
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    $\begingroup$ @CaptainLama There are various notational conventions for permutations. Here certainly the one-line notation is used. See en.wikipedia.org/wiki/Permutation. $\endgroup$ – Paul Frost Apr 29 at 16:56

The notation Wolfram is using isn’t cycle notation; see Paul Frost’s comment. The group $G$ they give is, using cycle notation, the group

$$\{e, (1~2), (3~4), (1~2)(3~4)\}.$$

Hopefully this makes it a bit more clear why the orbits are what they are.

  • $\begingroup$ I wouldn't say odd. See my above comment. But Wolfram should have made explicit which notation is used. $\endgroup$ – Paul Frost Apr 29 at 17:01
  • $\begingroup$ @PaulFrost Huh, I didn’t know people used one-line notation “in the wild,” so to speak. Thanks for the heads up. $\endgroup$ – Santana Afton Apr 29 at 17:09
  • $\begingroup$ @PaulFrost I would call it odd. Except in very specific cases, the cycle notation is what pretty much what everyone will expect. $\endgroup$ – Tobias Kildetoft Apr 29 at 17:26

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