A group action begins with a promise: a group $G$ moves a set $X$ around while respecting multiplication in $G$. The first question is not only where a point goes, but how much of the group fails to move it at all. Those unmoving elements carry information. They measure symmetry left over after choosing a point, and they explain why orbits often have sizes smaller than the group itself.

The stabiliser is the answer to that question. It turns an action into a family of subgroups, one subgroup for each point of the set being acted on. When the action comes from geometry, the stabiliser is the symmetry group of a chosen object. When the action comes from algebra, it records conjugation symmetries, fixed vectors, and automorphisms preserving extra structure.

[example: Rotations of a Square and a Chosen Vertex] Let $G$ be the rotational symmetry group of a square, so \begin{align*} G=C_4=\{e,r,r^2,r^3\}, \end{align*} where $r$ is rotation by $90$ degrees counterclockwise and $r^4=e$. Label the vertices cyclically by $x_0,x_1,x_2,x_3$, and choose $x=x_0$. Then \begin{align*} e\cdot x_0 &= x_0, \\ r\cdot x_0 &= x_1, \\ r^2\cdot x_0 &= x_2, \\ r^3\cdot x_0 &= x_3. \end{align*} Hence the orbit is \begin{align*} G\cdot x_0 &= \{g\cdot x_0 : g\in G\} \\ &= \{e\cdot x_0,r\cdot x_0,r^2\cdot x_0,r^3\cdot x_0\} \\ &= \{x_0,x_1,x_2,x_3\} = X. \end{align*} For the stabiliser, we test the same four elements: \begin{align*} e\cdot x_0 &= x_0, \\ r\cdot x_0 &= x_1 \ne x_0, \\ r^2\cdot x_0 &= x_2 \ne x_0, \\ r^3\cdot x_0 &= x_3 \ne x_0. \end{align*} Therefore \begin{align*} G_{x_0} &= \{g\in G : g\cdot x_0=x_0\} \\ &= \{e\}. \end{align*} Now let $G$ act on the set $Y$ of the two diagonals. Write \begin{align*} d_1 &= \{x_0,x_2\}, & d_2 &= \{x_1,x_3\}. \end{align*} The action on $d_1$ is \begin{align*} e\cdot d_1 &= \{e\cdot x_0,e\cdot x_2\}=\{x_0,x_2\}=d_1, \\ r\cdot d_1 &= \{r\cdot x_0,r\cdot x_2\}=\{x_1,x_3\}=d_2, \\ r^2\cdot d_1 &= \{r^2\cdot x_0,r^2\cdot x_2\}=\{x_2,x_0\}=d_1, \\ r^3\cdot d_1 &= \{r^3\cdot x_0,r^3\cdot x_2\}=\{x_3,x_1\}=d_2. \end{align*} Thus \begin{align*} G_{d_1} &= \{g\in G : g\cdot d_1=d_1\} \\ &= \{e,r^2\}. \end{align*} The chosen vertex has no residual rotational symmetry, while the chosen diagonal is still preserved by the half-turn $r^2$. [/example]

The example shows the basic tension. The orbit tells us how many positions a chosen object can occupy; the stabiliser tells us which group elements are invisible from that object's point of view. The main theorem of the subject says that these two measurements multiply to the size of the acting group when $G$ is finite.

## Definition

Before naming the stabiliser, we need the action itself to be part of the data. A group does not act merely because its elements look like transformations; we need a specified rule telling each group element how it moves each point, and that rule must respect the group law.

[definition: Group Action] Let $G$ be a group and let $X$ be a set. A group action of $G$ on $X$ is a map \begin{align*} G \times X &\to X \\ (g,x) &\mapsto g \cdot x \end{align*} such that for all $g,h \in G$ and all $x \in X$, \begin{align*} e \cdot x &= x, \\ g \cdot (h \cdot x) &= (gh) \cdot x, \end{align*} where $e$ is the identity element of $G$. [/definition]

Once the action is fixed, it becomes meaningful to ask which group elements become undetectable at a chosen point. We do not ask which elements are the identity in $G$; we ask which elements behave like the identity on that particular point.

[definition: Stabiliser] Let $G$ be a group acting on a set $X$, and let $x \in X$. The stabiliser of $x$ in $G$ is \begin{align*} G_x = \{g \in G : g \cdot x = x\}. \end{align*} [/definition]

For stabilisers to be useful in group theory, they must inherit the algebraic structure of $G$ rather than remain arbitrary subsets. The possible obstruction is that two elements may each fix $x$ for unrelated reasons, so it is not automatic from the definition alone that their product still fixes $x$, or that undoing one of them still fixes $x$. The action axioms are exactly what remove this obstruction.

[quotetheorem:5000]

This theorem converts geometric or combinatorial questions about points into algebraic questions about subgroups. To use it for counting, we also need the complementary object: the set of all positions that the point can reach under the group action.

[definition: Orbit] Let $G$ be a group acting on a set $X$, and let $x \in X$. The orbit of $x$ is \begin{align*} G \cdot x = \{g \cdot x : g \in G\}. \end{align*} [/definition]

The orbit is a subset of $X$, while the stabiliser is a subgroup of $G$. The two live in different places, but they are linked by a simple idea: two group elements send $x$ to the same point exactly when they differ by an element that fixes $x$.

## Orbits and Cosets

### The Orbit-Stabiliser Count

The stabiliser first pays off when we count. If $G$ is finite, a point cannot usually have $|G|$ distinct images, because several elements of $G$ may produce the same image. The stabiliser measures exactly how much repetition occurs.

To make that repetition visible, compare the map $g \mapsto g \cdot x$ with cosets of $G_x$. Elements in the same left coset of $G_x$ send $x$ to the same place, and elements in different left cosets send $x$ to different places.

[quotetheorem:845]

This is the main structural statement about stabilisers in finite group actions. It says that a point with a large stabiliser has a small orbit, and a point with a small stabiliser has a large orbit. The theorem also explains why orbit sizes must divide $|G|$.