Key Concepts of Group Actions

Why This Matters

Group actions are one of the most powerful unifying ideas in abstract algebra—they're how we formalize the notion that a group "does something" to a mathematical object. When you're tested on this material, you're not just being asked to recite definitions. You're being evaluated on whether you understand how orbits, stabilizers, fixed points, and counting formulas work together to reveal the structure of both the group and the set it acts on.

These concepts connect directly to symmetry analysis, counting problems, and the deep relationship between abstract groups and concrete permutations. The orbit-stabilizer theorem, Burnside's lemma, and Cayley's theorem appear repeatedly in proofs and applications. Don't just memorize the formulas—know which concept each result illustrates and when to apply each tool.

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Foundations: What Is a Group Action?

Before diving into theorems, you need a rock-solid understanding of what it means for a group to act on a set. A group action translates abstract group elements into concrete transformations.

Definition of a Group Action

• A group action of $G$ on a set $X$—formally, a map $G \times X \to X$ written $(g, x) \mapsto g \cdot x$ that respects group structure
• Two axioms must hold: the identity $e \cdot x = x$ for all $x$, and compatibility $(gh) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$
• Think of it as symmetries—each group element corresponds to a way of "rearranging" the set while preserving some structure

Group Actions on Sets

• Actions can be defined on any set—finite sets, infinite sets, vector spaces, geometric figures, or even the group itself
• The set's structure constrains the action—actions on finite sets yield permutation representations; actions on geometric objects reveal spatial symmetries
• This generality makes group actions ubiquitous—they appear in combinatorics, geometry, topology, and physics

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Orbits and Stabilizers: The Core Duality

The orbit-stabilizer relationship is the heart of group action theory. These two concepts are inversely related: larger stabilizers mean smaller orbits, and vice versa.

Orbits and Stabilizers

• The orbit of $x \in X$ is $\text{Orb}(x) = \{g \cdot x : g \in G\}$—all points reachable from $x$ by the group action
• The stabilizer of $x$ is $\text{Stab}(x) = \{g \in G : g \cdot x = x\}$—the subgroup of elements that fix $x$
• Orbits partition $X$ into disjoint subsets, so every element belongs to exactly one orbit

Orbit-Stabilizer Theorem

• The fundamental counting result: $|\text{Orb}(x)| = [G : \text{Stab}(x)] = \frac{|G|}{|\text{Stab}(x)|}$ for finite groups
• Connects group structure to action geometry—the orbit size equals the index of the stabilizer subgroup
• Essential for counting arguments—if you know any two of $|G|$, $|\text{Orb}(x)|$, or $|\text{Stab}(x)|$, you can find the third

Fixed Points and the Class Equation

• Fixed points satisfy $g \cdot x = x$ for all $g \in G$—they form singleton orbits
• The class equation for $G$ acting on itself by conjugation: $|G| = |Z(G)| + \sum [G : C_G(g_i)]$, summing over non-central conjugacy class representatives
• Powerful for proving structure theorems—the class equation shows that $p$-groups have nontrivial centers

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Special Types of Actions

Not all group actions are created equal. Transitive and regular actions represent extremes of symmetry and efficiency.

Transitive Actions

• A transitive action has exactly one orbit—for any $x, y \in X$, there exists $g \in G$ with $g \cdot x = y$
• Implies maximum "reachability"—the group can move any element to any other element
• Models homogeneous spaces—in geometry, a space is homogeneous if its symmetry group acts transitively (every point "looks the same")

Regular Actions

• Regular = transitive + free—exactly one orbit, and only the identity fixes any point ($\text{Stab}(x) = \{e\}$ for all $x$)
• By orbit-stabilizer: $|X| = |G|$ for regular actions, so the set has the same cardinality as the group
• The "cleanest" type of action—no redundancy, each group element corresponds to a unique permutation with no fixed points (except identity)

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Representation and Counting Tools

Group actions provide concrete ways to study abstract groups and solve counting problems. Every group is secretly a permutation group, and symmetry simplifies enumeration.

Cayley's Theorem

• Every group $G$ embeds into $S_{|G|}$—the symmetric group on $|G|$ elements, via the left regular representation
• Proof uses the action of $G$ on itself—$g$ maps to the permutation $\sigma_g(h) = gh$, which is faithful (injective)
• Foundational for group theory—shows that abstract groups are "just" permutation groups in disguise

Permutation Representations

• Any group action on a finite set $X$ yields a homomorphism $\phi: G \to S_X$, sending $g$ to the permutation it induces
• The kernel is the set of elements acting trivially—a faithful action means $\ker(\phi) = \{e\}$
• Connects algebra to combinatorics—group properties (order, structure) translate to permutation properties (cycle type, parity)

Burnside's Lemma

• Counts distinct orbits via the formula: $|\text{Orbits}| = \frac{1}{|G|} \sum_{g \in G} |X^g|$, where $X^g$ is the set of points fixed by $g$
• "Average the fixed points"—the number of orbits equals the average number of fixed points per group element
• Essential for combinatorial enumeration—counting distinct necklaces, colorings, or molecular configurations up to symmetry

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Quick Reference Table

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Self-Check Questions

1. If a group $G$ of order 12 acts on a set and some element $x$ has an orbit of size 4, what is the order of $\text{Stab}(x)$? Which theorem tells you this?

2. Compare transitive and regular actions: what property must a transitive action have to also be regular? Give an example of a transitive action that is not regular.

3. How does Burnside's lemma differ from the orbit-stabilizer theorem in application? If an FRQ asks "how many distinct colorings exist under rotational symmetry," which would you use?

4. Explain why Cayley's theorem guarantees that every group of order $n$ can be viewed as a subgroup of $S_n$. What group action makes this work?

5. The class equation includes the term $|Z(G)|$. What do elements of $Z(G)$ correspond to in terms of the conjugation action, and why do they appear separately in the equation?