What is this $\Sigma_4 /\langle (12)(34)\rangle$?

The question that I am supposed to answer is

Are the orbits $\Sigma_4 /\langle (12)(34)\rangle$ and $\Sigma_4 /\langle (13)(24)\rangle$ isomorphic as $\Sigma_4$ sets?

I do not understand what $\Sigma_4 /\langle (12)(34)\rangle$ is in this contex.

Definition of orbit that I know is:

If $x \in X$ (where $X$ is a set) and $G$ is a group then $\operatorname{orb} (x) = \{ \varphi_g(x) : g \in G \}$

So I understand that I can have orbit of an element but what is this $\Sigma_4 /\langle(12)(34)\rangle$ then?

Solution 1:

Assuming $\Sigma_4$ is the symmetric group on four symbols, the notation

$$\Sigma_4 /\langle (12)(34)\rangle$$

means the set of $\Sigma_4$ left cosets of the subgroup $\langle (12)(34)\rangle$ of $\Sigma_4$ generated by the element $(12)(34)$.

For a more specific definition involving actions, see this Wikipedia article:

The set of all orbits of $X$ under the action of $G$ is written as X/G (or, less frequently: G\X), and is called the quotient of the action.