Key Concepts of Cayley's Theorem to Know for Groups and Geometries

Cayley's Theorem shows that every group can be represented as a subgroup of a symmetric group, linking abstract group structures to concrete actions on sets. This connection helps us understand group properties through permutations and regular actions.

1. Statement of Cayley's Theorem

   • Every group ( G ) is isomorphic to a subgroup of the symmetric group ( S_G ).
   • This means that any group can be represented as a group of permutations.
   • The theorem highlights the connection between abstract groups and concrete actions on sets.

2. Definition of a regular group action

   • A group action is regular if it acts freely and transitively on a set.
   • For every element in the set, there is a unique group element that maps it to another element.
   • Regular actions ensure that the group structure can be fully realized through its action on the set.

3. Concept of group isomorphism

   • An isomorphism is a bijective homomorphism between two groups, preserving the group operation.
   • If two groups are isomorphic, they have the same structure, even if their elements differ.
   • Isomorphisms allow us to classify groups based on their structural properties rather than their specific elements.

4. Symmetric group and its properties

   • The symmetric group ( S_n ) consists of all permutations of ( n ) elements.
   • It has ( n! ) elements, making it one of the largest groups for a given ( n ).
   • The symmetric group is non-abelian for ( n \geq 3 ) and serves as a fundamental example in group theory.

5. Left regular representation

   • The left regular representation of a group ( G ) is a way to represent ( G ) as a group of permutations on itself.
   • Each group element corresponds to a permutation that shifts elements of the group.
   • This representation is crucial for demonstrating Cayley's Theorem, as it shows how ( G ) can act on itself.

6. Proof outline of Cayley's Theorem

   • Define a function ( \phi: G \to S_G ) that maps each element ( g \in G ) to the permutation it induces.
   • Show that ( \phi ) is a homomorphism and is injective.
   • Conclude that the image of ( \phi ) is a subgroup of ( S_G ), establishing the isomorphism.

7. Implications for finite groups

   • Cayley's Theorem implies that every finite group can be represented as a permutation group.
   • This representation aids in understanding the structure and properties of finite groups.
   • It also allows for the application of combinatorial techniques to study group properties.

8. Applications in group theory

   • Cayley's Theorem is foundational for studying group actions and representations.
   • It provides a method for visualizing abstract groups through permutations.
   • The theorem is used in various areas, including algebra, geometry, and combinatorics.

9. Relationship to permutation groups

   • Permutation groups are a specific type of group that consists of all bijective functions on a set.
   • Cayley's Theorem shows that any group can be viewed as a permutation group, linking abstract algebra to combinatorial structures.
   • This relationship is essential for understanding the symmetry and structure of mathematical objects.

10. Examples illustrating Cayley's Theorem

    • The group ( \mathbb{Z}/3\mathbb{Z} ) can be represented as a subgroup of ( S_3 ) through its action on three elements.
    • The symmetric group ( S_4 ) can illustrate how a larger group can contain smaller groups as subgroups.
    • The dihedral group ( D_4 ) can be shown to act on the vertices of a square, demonstrating the theorem's applicability to geometric symmetries.