All Study Guides Groups and Geometries Unit 4
⭕ Groups and Geometries Unit 4 – Group Actions and SymmetryGroup actions provide a powerful framework for understanding symmetry in mathematics. They formalize how groups transform sets, revealing underlying structures and relationships. This concept bridges abstract algebra and geometry, offering insights into the symmetries of objects and spaces.
Orbits, stabilizers, and different types of actions are key tools for analyzing group actions. These concepts help classify symmetries, count distinct configurations, and solve problems in various mathematical fields, from geometry to number theory and beyond.
Key Concepts and Definitions
Group action involves a group G G G acting on a set X X X through a function ϕ : G × X → X \phi: G \times X \to X ϕ : G × X → X that satisfies certain conditions
Orbit of an element x ∈ X x \in X x ∈ X under the action of G G G is the set O r b G ( x ) = { g ⋅ x : g ∈ G } Orb_G(x) = \{g \cdot x : g \in G\} O r b G ( x ) = { g ⋅ x : g ∈ G }
Elements in the same orbit are considered equivalent under the group action
Stabilizer of an element x ∈ X x \in X x ∈ X is the subgroup S t a b G ( x ) = { g ∈ G : g ⋅ x = x } Stab_G(x) = \{g \in G : g \cdot x = x\} St a b G ( x ) = { g ∈ G : g ⋅ x = x }
Consists of all group elements that fix x x x under the action
Transitive action occurs when there is only one orbit, meaning any two elements in X X X can be mapped to each other by some group element
Faithful action implies that the only group element that fixes every element of X X X is the identity element
Symmetry group of an object is the group of all transformations that leave the object unchanged (isometries)
Examples include rotation, reflection, and translation groups
Group Theory Foundations
Groups are algebraic structures consisting of a set G G G and a binary operation ∗ * ∗ satisfying closure, associativity, identity, and inverse properties
Subgroups are subsets of a group that form a group under the same operation
Lagrange's Theorem states that the order of a subgroup divides the order of the group
Cosets are sets of the form a H = { a h : h ∈ H } aH = \{ah : h \in H\} a H = { ah : h ∈ H } for a subgroup H H H and an element a ∈ G a \in G a ∈ G
Left and right cosets partition the group into disjoint sets of equal size
Normal subgroups are subgroups N N N such that a N = N a aN = Na a N = N a for all a ∈ G a \in G a ∈ G
Allow for the construction of quotient groups G / N G/N G / N
Homomorphisms are structure-preserving maps between groups that respect the group operation
Isomorphisms are bijective homomorphisms, indicating that two groups have the same structure
Introduction to Group Actions
Group actions formalize the idea of a group G G G transforming the elements of a set X X X
The action is defined by a function ϕ : G × X → X \phi: G \times X \to X ϕ : G × X → X satisfying:
ϕ ( e , x ) = x \phi(e, x) = x ϕ ( e , x ) = x for all x ∈ X x \in X x ∈ X , where e e e is the identity element of G G G
ϕ ( g 1 , ϕ ( g 2 , x ) ) = ϕ ( g 1 g 2 , x ) \phi(g_1, \phi(g_2, x)) = \phi(g_1g_2, x) ϕ ( g 1 , ϕ ( g 2 , x )) = ϕ ( g 1 g 2 , x ) for all g 1 , g 2 ∈ G g_1, g_2 \in G g 1 , g 2 ∈ G and x ∈ X x \in X x ∈ X
The notation g ⋅ x g \cdot x g ⋅ x is often used to denote ϕ ( g , x ) \phi(g, x) ϕ ( g , x )
Group actions can be thought of as permutations of the set X X X induced by the elements of G G G
The set X X X is called a G G G -set when there is a group action of G G G on X X X
Group actions provide a way to study the symmetries and structure of mathematical objects
Types of Group Actions
Transitive action: For any x , y ∈ X x, y \in X x , y ∈ X , there exists a g ∈ G g \in G g ∈ G such that g ⋅ x = y g \cdot x = y g ⋅ x = y
Equivalently, there is only one orbit under the action
Faithful action: For any g ∈ G g \in G g ∈ G , if g ⋅ x = x g \cdot x = x g ⋅ x = x for all x ∈ X x \in X x ∈ X , then g = e g = e g = e
The action is injective, and G G G can be embedded into the symmetric group S y m ( X ) Sym(X) S y m ( X )
Free action: For any x ∈ X x \in X x ∈ X and g ∈ G g \in G g ∈ G , if g ⋅ x = x g \cdot x = x g ⋅ x = x , then g = e g = e g = e
Stabilizers are trivial, and each group element moves every point
Regular action: The action is both transitive and free
G G G acts on itself by left multiplication, g ⋅ h = g h g \cdot h = gh g ⋅ h = g h
Trivial action: For all g ∈ G g \in G g ∈ G and x ∈ X x \in X x ∈ X , g ⋅ x = x g \cdot x = x g ⋅ x = x
Every group element fixes every point in X X X
Orbits and Stabilizers
The orbit of an element x ∈ X x \in X x ∈ X under the action of G G G is the set O r b G ( x ) = { g ⋅ x : g ∈ G } Orb_G(x) = \{g \cdot x : g \in G\} O r b G ( x ) = { g ⋅ x : g ∈ G }
Orbits partition X X X into disjoint subsets
The stabilizer of an element x ∈ X x \in X x ∈ X is the subgroup S t a b G ( x ) = { g ∈ G : g ⋅ x = x } Stab_G(x) = \{g \in G : g \cdot x = x\} St a b G ( x ) = { g ∈ G : g ⋅ x = x }
Measures the symmetry of x x x under the action
The Orbit-Stabilizer Theorem states that for any x ∈ X x \in X x ∈ X , ∣ O r b G ( x ) ∣ ⋅ ∣ S t a b G ( x ) ∣ = ∣ G ∣ |Orb_G(x)| \cdot |Stab_G(x)| = |G| ∣ O r b G ( x ) ∣ ⋅ ∣ St a b G ( x ) ∣ = ∣ G ∣
Relates the sizes of orbits and stabilizers to the size of the group
Burnside's Lemma counts the number of orbits under a group action
∣ X / G ∣ = 1 ∣ G ∣ ∑ g ∈ G ∣ F i x ( g ) ∣ |X/G| = \frac{1}{|G|} \sum_{g \in G} |Fix(g)| ∣ X / G ∣ = ∣ G ∣ 1 ∑ g ∈ G ∣ F i x ( g ) ∣ , where F i x ( g ) = { x ∈ X : g ⋅ x = x } Fix(g) = \{x \in X : g \cdot x = x\} F i x ( g ) = { x ∈ X : g ⋅ x = x }
Symmetry Groups
Symmetry groups capture the symmetries of geometric objects or patterns
The symmetry group of an object consists of all transformations that leave the object unchanged (isometries)
Common symmetry groups include:
Cyclic groups C n C_n C n for rotational symmetry
Dihedral groups D n D_n D n for rotational and reflectional symmetry
Frieze groups for patterns with translational symmetry in one direction
Wallpaper groups for patterns with translational symmetry in two directions
Studying the symmetry group of an object reveals its underlying structure and properties
For example, the symmetry group of a regular polygon is the dihedral group D n D_n D n
Applications in Geometry
Group actions are used to study the symmetries of geometric objects and spaces
Isometry groups are symmetry groups that preserve distances (rigid motions)
Examples include the Euclidean group E ( n ) E(n) E ( n ) and the orthogonal group O ( n ) O(n) O ( n )
Lie groups are continuous symmetry groups that play a crucial role in physics and geometry
Examples include the special orthogonal group S O ( n ) SO(n) SO ( n ) and the unitary group U ( n ) U(n) U ( n )
Orbifolds are quotient spaces obtained by identifying points that are equivalent under a group action
Provide a way to study the geometry and topology of spaces with symmetries
Group actions on manifolds and algebraic varieties are used to classify and understand their structure
Quotient spaces, fixed points, and orbit spaces are important tools in this context
Problem-Solving Techniques
Identify the group G G G , the set X X X , and the action ϕ : G × X → X \phi: G \times X \to X ϕ : G × X → X
Verify that the action satisfies the required properties
Determine the orbits of elements in X X X under the action
Use the definition O r b G ( x ) = { g ⋅ x : g ∈ G } Orb_G(x) = \{g \cdot x : g \in G\} O r b G ( x ) = { g ⋅ x : g ∈ G }
Find the stabilizers of elements in X X X
Use the definition S t a b G ( x ) = { g ∈ G : g ⋅ x = x } Stab_G(x) = \{g \in G : g \cdot x = x\} St a b G ( x ) = { g ∈ G : g ⋅ x = x }
Apply the Orbit-Stabilizer Theorem to relate the sizes of orbits and stabilizers
∣ O r b G ( x ) ∣ ⋅ ∣ S t a b G ( x ) ∣ = ∣ G ∣ |Orb_G(x)| \cdot |Stab_G(x)| = |G| ∣ O r b G ( x ) ∣ ⋅ ∣ St a b G ( x ) ∣ = ∣ G ∣
Use Burnside's Lemma to count the number of orbits
∣ X / G ∣ = 1 ∣ G ∣ ∑ g ∈ G ∣ F i x ( g ) ∣ |X/G| = \frac{1}{|G|} \sum_{g \in G} |Fix(g)| ∣ X / G ∣ = ∣ G ∣ 1 ∑ g ∈ G ∣ F i x ( g ) ∣
Identify the type of action (transitive, faithful, free, regular, trivial)
Use the definitions and properties of each type to classify the action
Exploit the symmetries of the object or space under consideration
Determine the symmetry group and use its properties to simplify the problem