Key Concepts of Group Actions to Know for Groups and Geometries

Group actions describe how a group can operate on a set, revealing symmetries and relationships between the two. Key concepts include orbits, stabilizers, and the orbit-stabilizer theorem, which help us understand the structure of groups and their actions.

1. Definition of a group action

   • A group action is a formal way for a group to operate on a set, associating each group element with a function that maps elements of the set to itself.
   • It must satisfy two properties: the identity element acts as the identity on the set, and the action of the product of two group elements is the composition of their actions.
   • Group actions can be thought of as symmetries of the set, revealing how the structure of the group relates to the set.

2. Orbits and stabilizers

   • The orbit of an element is the set of all points in the set that can be reached by the action of the group on that element.
   • The stabilizer of an element is the subgroup of the group that keeps that element fixed under the group action.
   • Orbits partition the set into disjoint subsets, while stabilizers provide insight into the symmetry of individual elements.

3. Orbit-stabilizer theorem

   • This theorem connects the size of the orbit of an element with the size of its stabilizer, stating that the size of the orbit is equal to the index of the stabilizer in the group.
   • It provides a powerful tool for counting the number of distinct elements in an orbit.
   • The theorem emphasizes the relationship between group actions and subgroup structures.

4. Transitive actions

   • A group action is transitive if there is only one orbit, meaning any element can be reached from any other element by the action of some group element.
   • Transitive actions imply a high degree of symmetry within the set.
   • They are often used to study homogeneous spaces in geometry.

5. Regular actions

   • A group action is regular if it is both transitive and free, meaning that the only group element that fixes any point is the identity.
   • Regular actions correspond to the idea of a group acting like a permutation group on a set.
   • They provide a clear structure for understanding how groups can act without redundancy.

6. Cayley's theorem

   • This theorem states that every group can be represented as a group of permutations, specifically as a subgroup of the symmetric group on its elements.
   • It shows that group actions can be understood through permutations, linking abstract algebra with combinatorial structures.
   • Cayley's theorem is foundational for understanding the relationship between groups and their actions.

7. Permutation representations

   • A permutation representation is a way of representing a group as a group of permutations on a set, often derived from a group action.
   • This representation allows for the study of group properties through the lens of combinatorial objects.
   • It highlights the connection between algebraic structures and symmetric properties.

8. Fixed points and the class equation

   • Fixed points are elements of the set that remain unchanged under the action of the group.
   • The class equation relates the size of a group to the sizes of its conjugacy classes and the number of fixed points, providing insight into the structure of the group.
   • Understanding fixed points is crucial for analyzing the behavior of group actions.

9. Group actions on sets

   • Group actions can be defined on various types of sets, including finite sets, infinite sets, and geometric objects.
   • The nature of the set influences the properties of the group action, such as transitivity and regularity.
   • Group actions on sets are fundamental in many areas of mathematics, including geometry and combinatorics.

10. Burnside's lemma

    • Burnside's lemma provides a way to count the number of distinct orbits of a group action by averaging the number of points fixed by each group element.
    • It is particularly useful in combinatorial enumeration problems, such as counting distinct colorings or arrangements.
    • The lemma highlights the interplay between symmetry and counting, making it a powerful tool in group theory.