# Group orbit: confusion

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 $$\{3,4\}$$.

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.

• It's simple: $G$ is not a group. For instance $(1234)^2=(13)(24)$ is not in $G$. – Captain Lama Apr 29 at 16:47
• I think you made a confusion with the double-transposition group. – Captain Lama Apr 29 at 16:48
• 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. – Ned Apr 29 at 16:52
• @CaptainLama There are various notational conventions for permutations. Here certainly the one-line notation is used. See en.wikipedia.org/wiki/Permutation. – 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
$$\{e, (1~2), (3~4), (1~2)(3~4)\}.$$