Orbit of the Little Group 
I have a problem with the comprehension of the definition of an orbit.
Right now I'm writing my bachelor thesis about the analysis of solids with help of group theory and I need to understand this definition in order to explain how to get an irreducible representation of symmorphic space groups.
First I will give some background for notations etc..
Let the Group $\mathcal{G}$ be a semi-direct product:
$\mathcal{G} = \mathcal{A}\rtimes\mathcal{B}$
where $\mathcal{A}$ is an abelian, invariant subgroup of $\mathcal{G}$ and $\mathcal{B}$ a subgroup of $\mathcal{G}$.
The little group is defined as:
The subset $\mathcal{B}$(q) ⊂ $\mathcal{B}$ containing all elements
B ∈ $\mathcal{B}$ with the property:
$$\chi^q_\mathcal{A}(BAB^{-1})=\chi^q_\mathcal{A}(A)\hspace{1cm}      \text{for all} A ∈ \mathcal{A} \tag{1}$$
is called the little group of q (with respect to $\mathcal{A}$).
$\chi^q_\mathcal{A}$ denotes the character of qth irreducible representation of $\mathcal{A}$.
The book which I read for the theory provides the following definition for the orbit:
"Because $\mathcal{B}(q) ⊂ \mathcal{B}$ it's possible to obtain $M(q) =$ ord($\mathcal{B}$)/ord($\mathcal{B}$(q)) right cosets of $\mathcal{B}$ with respect to $\mathcal{B}$(q)."
So far so clear, but I don't get the next sentence:\

In a similar manner to $(eq1)$, it is possible to define a map $\mathcal{B}(q)$ of the integers
$q = 1, … , a$  to itself, by fixing a $B ∈ \mathcal{B}$ and writing:
$$\chi^{B(q)}_\mathcal{A}=\chi^q_\mathcal{A}(BAB^{-1}) \tag{2} $$

I don't get the idea of the map of integers to itself... They represent the $q$th irr. Rep.. What is the connection to $(eq2)$?
Then there follows a calculation, which I skip. In the end it states:
$\chi^{B(q)}_\mathcal{A}=\chi^{B_j(q)}_\mathcal{A}(A)$
where $B_j(q)$ denotes the $j$th of $M(q)$ coset representive for the right cosets of $\mathcal{B}$ with respect to $\mathcal{B}(q)$.
Finally it follows the definition of the orbit of $q$:
The set of $M(q)$ integers {
$B_1(q), … , B_{M(q)}(q)$
}
is called the orbit of $q$.
Clearly my problem is connected to the map of these integers.
I don't see the connection between those character formulas and the number of the irr.Rep.s.
Maybe some of you can follow the idea of the author and can share it with me or know a different source where this definition is explained in a different way.
 A: Since $\mathcal A$ is abelian, all irreducible representations are 1-dimensional and form the space $X=\text{Hom}(\mathcal A,\mathbb C^\times)$. Since $\mathcal A$ is a normal subgroup, the group $\mathcal G=\mathcal B\ltimes A$ (or subgroup $\mathcal B$ if you want) has an action on $X$ by
$$ (g\cdot\chi)(a) = \chi(gag^{-1}),$$
where $g\in\mathcal G$, $a\in\mathcal A$ and $\chi\in X$. If $\chi$ is an irreducible representation of $A$, then so is $g\cdot\chi\in X$. Having a group action, you can define stabilizer subgroups (little group) and orbits
$$ \text{Orb}(\chi) = \{g\cdot\chi\in X | \forall g\in\mathcal G\}.$$
We can write the above in a slightly different way by labeling the set of irreducible representations of $A$ by integers $q$,
$$ \left\{\chi_{q}\right\}.$$
Since for any character $\chi_{q}$, the action $g\cdot\chi_q$ is another character is must be equal to
$$g\cdot\chi_q = \chi_{q'},$$
for some other integer $q'$. Since the action permutes the characters, we can think of it as effectively acting on the integer labeling the character
$$ g\cdot\chi_q = \chi_{g(q)}. $$
Although I don't think the label $q$ are integers, unless $\mathcal A$ is a finite group.
In representations of symmorphic space groups (in physics), $\mathcal A$ is the group of translations (Bravais lattice) and its irreducible unitary representations are classified by the Brillouin torus
$$ T_B = \mathbb R^d/\mathcal A,$$
the choice of representative of which are usually called Brillouin zones. Elements of $T_B$ are called wave vectors . For any $q\in T_B$ we have a representation of $\mathcal A$ (Bloch wave functions in physics litterature). The action of the space group $\mathcal G$ (or point group $\mathcal B$) on characters/bloch functions $\chi_q$ permutes the characters
$$ g\cdot\chi_q = \chi_{g(q)}. $$
Thus if a wave vector $q$ appears in a representation of the space group $\mathcal G$, then all other wave vectors in its orbit must also appear (as they are permuted amongst each other by elements in $\mathcal G$).
I hope this answered your question somewhat. And by the way, can I ask which book are you using?
