Group Actions | Mathematics - Wyatt's Notes

A group action of $G$ on a set $X$ is a map $G \times X \to X$Written $(g, x) \mapsto g \cdot x$ Satisfying:

1. $e \cdot x = x$ for all $x \in X$.
2. $g \cdot (h \cdot x) = (gh) \cdot x$ for all $g, h \in G$ and $x \in X$.

The orbit of $x \in X$ is $\mathrm{Orb}(x) = \{g \cdot x : g \in G\}$.

The stabilizer of $x \in X$ is $\mathrm{Stab}(x) = \{g \in G : g \cdot x = x\}$.

Proposition 6.1. $\mathrm{Stab}(x)$ is a subgroup of $G$.

Theorem 6.2 (Orbit-Stabilizer Theorem). For any $x \in X$

$|\mathrm{Orb}(x)| = [G : \mathrm{Stab}(x)] = \frac{|G|}{|\mathrm{Stab}(x)|}$

Proof. Define $\phi : G \to \mathrm{Orb}(x)$ by $\phi(g) = g \cdot x$. Then $g$ and $h$ have the same Image iff $g \cdot x = h \cdot x$ iff $h^{-1}g \cdot x = x$ iff $h^{-1}g \in \mathrm{Stab}(x)$ Iff $g \in h\,\mathrm{Stab}(x)$. So the fibers of $\phi$ are precisely the cosets of $\mathrm{Stab}(x)$ And there are $[G : \mathrm{Stab}(x)]$ of them, each mapping to a distinct element of $\mathrm{Orb}(x)$. $\blacksquare$

Theorem 6.3 (Burnside’s Lemma). If a finite group $G$ acts on a finite set $X$ Then the number Of orbits is

$\frac{1}{|G|} \sum_{g \in G} |\mathrm{Fix}(g)|$

Where $\mathrm{Fix}(g) = \{x \in X : g \cdot x = x\}$.

Proof. Count the set $S = \{(g, x) \in G \times X : g \cdot x = x\}$ in two ways. Grouping by $g$: $|S| = \sum_{g \in G} |\mathrm{Fix}(g)|$. Grouping by $x$: $|S| = \sum_{x \in X} |\mathrm{Stab}(x)|$. For $x$ in orbit $O$, $|\mathrm{Stab}(x)| = |G|/|O|$. So $\sum_{x \in O} |\mathrm{Stab}(x)| = |O| \cdot |G|/|O| = |G|$. Summing over all orbits: $|S| = |G| \cdot (\mathrm{number\ of\ orbits})$. $\blacksquare$

$G$ acts on itself by conjugation: $g \cdot x = gxg^{-1}$.

The orbits are called conjugacy classes. The stabilizer of $x$ is the centralizer $C_G(x) = \{g \in G : gx = xg\}$.

Theorem 6.4 (Class Equation). For a finite group $G$

$|G| = |Z(G)| + \sum_{i} [G : C_G(x_i)]$

Where the sum is over representatives $x_i$ of the non-central conjugacy classes.

Proof. The conjugacy classes partition $G$. Central elements form singleton classes. For a non-central element $x$, $|\mathrm{Orb}(x)| = [G : C_G(x)]$ by the orbit-stabilizer theorem. Summing gives the result. $\blacksquare$

Problem. The rotational symmetry group of a cube has $24$ elements. Use the orbit-stabilizer theorem To verify the sizes of the orbits of vertices, edges, and faces under this action.

Theorem 6.5. If $G$ is a non-trivial finite $p$-group (i.e., $|G| = p^n$ for some prime $p$ and $n \geq 1$), then $Z(G)$ is non-trivial: $|Z(G)| \geq p$.

Proof. By the class equation:

$|G| = |Z(G)| + \sum_{i=1}^{r} [G : C_G(x_i)]$

Where $x_1, \ldots, x_r$ are representatives of the non-central conjugacy classes. For each $i$ $x_i$ is non-central, so $C_G(x_i) \neq G$. Thus $[G : C_G(x_i)]$ is a divisor of $|G| = p^n$ That is strictly greater than $1$Hence $p$ divides $[G : C_G(x_i)]$. Since $p$ also divides $|G|$ We have:

$|Z(G)| = |G| - \sum_{i=1}^{r} [G : C_G(x_i)] \equiv 0 - 0 \equiv 0 \pmod{p}$

Since $e \in Z(G)$We have $|Z(G)| \geq 1$. Therefore $|Z(G)| \geq p$. $\blacksquare$

Corollary 6.6. Every group of order $p^2$ (where $p$ is prime) is abelian.

Proof. By Theorem 6.5, $|Z(G)| \geq p$. Since $Z(G) \leq G$, $|Z(G)|$ divides $p^2$ So $|Z(G)| = p$ or $|Z(G)| = p^2$. If $|Z(G)| = p^2$ Then $G = Z(G)$ is abelian. If $|Z(G)| = p$ Then $G/Z(G)$ has order $p$ and is therefore cyclic, say $G/Z(G) = \langle gZ(G) \rangle$. Then every element of $G$ has the form $g^k z$ for some $k \in \mathbb{Z}$ and $z \in Z(G)$. For any two such elements $(g^{k_1}z_1)(g^{k_2}z_2) = g^{k_1+k_2}z_1z_2 = g^{k_2+k_1}z_2z_1 = (g^{k_2}z_2)(g^{k_1}z_1)$ So $G$ is abelian, contradicting $|Z(G)| = p$. Thus $|Z(G)| = p^2$ and $G$ is abelian. $\blacksquare$

flowchart TD
    A[6_Group Actions] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Group actions describe how symmetry groups interact with sets. A group acts on a set by assigning to each group element a permutation of the set, subject to compatibility conditions. The orbit of an element is the set of all positions reachable by the group action, while the stabiliser is the subgroup that fixes that element. The orbit-stabilizer theorem links these quantities: the size of the orbit equals the index of the stabiliser, relating local symmetry to global structure. Burnside’s lemma counts orbits by averaging fixed points, turning a potentially difficult enumeration into a manageable computation. These tools appear throughout mathematics, from counting colourings of objects to classifying crystal structures.

• Forgetting that the stabilizer is always a subgroup (it inherits identity and closure from the group axioms).
• Confusing the orbit of $x$ with the orbit of $G$: the orbit is a subset of $X$, not of $G$.
• Assuming that the number of orbits equals $|X|/|G|$; this is only true when the action is free.
• Misapplying Burnside’s lemma by using the wrong group action (e.g., using conjugation when the problem specifies a different action).
• Assuming that two elements in the same conjugacy class have the same centralizer; they have conjugate centralizers, but the sizes are equal.
• Confusing $[G : C_G(x)]$ (the index of the centralizer) with $|G|/|Z(G)|$ (the size of the quotient by the centre).

• The Sylow Theorems — Sylow’s theorems use group actions on coset spaces and conjugacy classes to count subgroups.
• Lagrange’s Theorem — The orbit-stabilizer theorem is a generalisation of Lagrange’s theorem applied to group actions.
• Classification of Groups of Small Order — Group actions on conjugacy classes and Sylow subgroups drive the classification of small-order groups.
• Worked Examples — Several worked examples apply orbit-stabilizer and Burnside’s lemma to concrete counting problems.

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.