Groups and Geometries

Groups and Geometries Unit 4 – Group Actions and Symmetry

Group 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 GG acting on a set XX through a function ϕ:G×XX\phi: G \times X \to X that satisfies certain conditions
  • Orbit of an element xXx \in X under the action of GG is the set OrbG(x)={gx:gG}Orb_G(x) = \{g \cdot x : g \in G\}
    • Elements in the same orbit are considered equivalent under the group action
  • Stabilizer of an element xXx \in X is the subgroup StabG(x)={gG:gx=x}Stab_G(x) = \{g \in G : g \cdot x = x\}
    • Consists of all group elements that fix xx under the action
  • Transitive action occurs when there is only one orbit, meaning any two elements in XX can be mapped to each other by some group element
  • Faithful action implies that the only group element that fixes every element of XX 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 GG 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 aH={ah:hH}aH = \{ah : h \in H\} for a subgroup HH and an element aGa \in G
    • Left and right cosets partition the group into disjoint sets of equal size
  • Normal subgroups are subgroups NN such that aN=NaaN = Na for all aGa \in G
    • Allow for the construction of quotient groups G/NG/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 GG transforming the elements of a set XX
  • The action is defined by a function ϕ:G×XX\phi: G \times X \to X satisfying:
    1. ϕ(e,x)=x\phi(e, x) = x for all xXx \in X, where ee is the identity element of GG
    2. ϕ(g1,ϕ(g2,x))=ϕ(g1g2,x)\phi(g_1, \phi(g_2, x)) = \phi(g_1g_2, x) for all g1,g2Gg_1, g_2 \in G and xXx \in X
  • The notation gxg \cdot x is often used to denote ϕ(g,x)\phi(g, x)
  • Group actions can be thought of as permutations of the set XX induced by the elements of GG
  • The set XX is called a GG-set when there is a group action of GG on XX
  • Group actions provide a way to study the symmetries and structure of mathematical objects

Types of Group Actions

  • Transitive action: For any x,yXx, y \in X, there exists a gGg \in G such that gx=yg \cdot x = y
    • Equivalently, there is only one orbit under the action
  • Faithful action: For any gGg \in G, if gx=xg \cdot x = x for all xXx \in X, then g=eg = e
    • The action is injective, and GG can be embedded into the symmetric group Sym(X)Sym(X)
  • Free action: For any xXx \in X and gGg \in G, if gx=xg \cdot x = x, then g=eg = e
    • Stabilizers are trivial, and each group element moves every point
  • Regular action: The action is both transitive and free
    • GG acts on itself by left multiplication, gh=ghg \cdot h = gh
  • Trivial action: For all gGg \in G and xXx \in X, gx=xg \cdot x = x
    • Every group element fixes every point in XX

Orbits and Stabilizers

  • The orbit of an element xXx \in X under the action of GG is the set OrbG(x)={gx:gG}Orb_G(x) = \{g \cdot x : g \in G\}
    • Orbits partition XX into disjoint subsets
  • The stabilizer of an element xXx \in X is the subgroup StabG(x)={gG:gx=x}Stab_G(x) = \{g \in G : g \cdot x = x\}
    • Measures the symmetry of xx under the action
  • The Orbit-Stabilizer Theorem states that for any xXx \in X, OrbG(x)StabG(x)=G|Orb_G(x)| \cdot |Stab_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=1GgGFix(g)|X/G| = \frac{1}{|G|} \sum_{g \in G} |Fix(g)|, where Fix(g)={xX:gx=x}Fix(g) = \{x \in X : g \cdot 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 CnC_n for rotational symmetry
    • Dihedral groups DnD_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 DnD_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) and the orthogonal group 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 SO(n)SO(n) and the unitary group 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 GG, the set XX, and the action ϕ:G×XX\phi: G \times X \to X
    • Verify that the action satisfies the required properties
  • Determine the orbits of elements in XX under the action
    • Use the definition OrbG(x)={gx:gG}Orb_G(x) = \{g \cdot x : g \in G\}
  • Find the stabilizers of elements in XX
    • Use the definition StabG(x)={gG:gx=x}Stab_G(x) = \{g \in G : g \cdot x = x\}
  • Apply the Orbit-Stabilizer Theorem to relate the sizes of orbits and stabilizers
    • OrbG(x)StabG(x)=G|Orb_G(x)| \cdot |Stab_G(x)| = |G|
  • Use Burnside's Lemma to count the number of orbits
    • X/G=1GgGFix(g)|X/G| = \frac{1}{|G|} \sum_{g \in G} |Fix(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


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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