Groups earn their keep by moving things: symmetries move the corners of a polygon, permutations move numbers, matrices move vectors. A group action makes "moving things" precise, and it turns many counting problems into group theory.
Definition. A group $G$ acts on a set $X$ if each $g \in G$ and $x \in X$ give a point $g \cdot x \in X$, with $$e \cdot x = x \quad \text{and} \quad g \cdot (h \cdot x) = (gh) \cdot x.$$ Equivalently, each $g$ gives a permutation of $X$, and $g \mapsto$ (that permutation) is a homomorphism from $G$ to the permutations of $X$.
- $S_{n}$ acts on $\{1, \dots, n\}$ by $\sigma \cdot i = \sigma(i)$.
- $D_{n}$ acts on the vertices, and on the edges, of a regular $n$-gon.
- $GL_{2}(\mathbb{R})$ acts on $\mathbb{R}^{2}$ by $A \cdot v = Av$.
- $G$ acts on itself by left multiplication, $g \cdot x = gx$ (Cayley's theorem), and by conjugation, $g \cdot x = gxg^{-1}$.
Orbits and stabilizers. The orbit of $x$ is $G \cdot x = \{g \cdot x : g \in G\}$, everywhere $x$ can be moved. The stabilizer of $x$ is $G_{x} = \{g \in G : g \cdot x = x\}$, the elements that leave it in place.
Why it works
(2) Each stabilizer is a subgroup. $e \cdot x = x$; if $g, h$ fix $x$ then $(gh) \cdot x = g \cdot (h \cdot x) = x$, and $g^{-1} \cdot x = g^{-1} \cdot (g \cdot x) = e \cdot x = x$. $\square$
Worked example: Orbits of a cyclic group
- Powers of $\sigma = (1\;2\;3)(4\;5)$ move $1, 2, 3$ around their cycle and swap $4, 5$; nothing moves $6$.
- Orbits: $\{1, 2, 3\}$, $\{4, 5\}$ and $\{6\}$.
Answer: Three orbits: the cycles of $\sigma$, with the fixed point as its own orbit.
Worked example: A stabilizer in $S_{4}$
- $\sigma$ fixes the set $\{1, 2\}$ when it sends $\{1, 2\}$ to itself, and hence $\{3, 4\}$ to itself.
- So $\sigma$ permutes $1, 2$ ($2$ ways) and $3, 4$ ($2$ ways): $\{e, (1\;2), (3\;4), (1\;2)(3\;4)\}$.
Answer: $4$ elements.
Common mistake
Key idea