1
$\begingroup$

Definition:

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?

Thanks.

$\endgroup$
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ I think you made a confusion with the double-transposition group. $\endgroup$ – Captain Lama Apr 29 at 16:48
  • 4
    $\begingroup$ 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
  • 1
    $\begingroup$ @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
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.