Abelian Groups

Definition

Abelian groups are mathematical structures that consist of a set equipped with an operation that combines any two elements to form a third element, following specific properties. They are characterized by the property that the group operation is commutative, meaning that the order in which two elements are combined does not affect the result. This property is crucial in understanding representations, particularly irreducible representations, as it influences how symmetries can be represented in a linear format and affects the construction and interpretation of character tables.

5 Must Know Facts For Your Next Test

1. In an abelian group, for any two elements 'a' and 'b', the equation 'a + b = b + a' holds true, making it fundamental for symmetry operations in representations.
2. Every finite abelian group can be expressed as a direct sum of cyclic groups, allowing for easier analysis of their structure.
3. The center of a group is always abelian; thus, abelian groups serve as a natural example of groups where every element commutes.
4. The character table of a finite group is particularly simplified when the group is abelian because all irreducible representations are one-dimensional.
5. In representation theory, the irreducibility of representations over abelian groups is guaranteed, as all their representations can be reduced to one-dimensional ones.

Review Questions

• How does the commutative property of abelian groups affect the study of irreducible representations?
  • The commutative property in abelian groups ensures that any two elements can be combined in any order without changing the result. This property simplifies the representation theory because it allows all representations of an abelian group to be reducible to one-dimensional representations. Thus, understanding how these groups operate helps in analyzing more complex structures and their irreducible representations.
• Discuss the implications of abelian groups on character tables and how they differ from non-abelian groups.
  • Abelian groups have character tables that are notably simpler than those of non-abelian groups. Since every irreducible representation of an abelian group is one-dimensional, the character table will consist of linear entries corresponding directly to group elements. In contrast, non-abelian groups have more complex character tables with higher dimensional representations and intricate relationships among characters due to non-commutativity.
• Evaluate how the properties of abelian groups contribute to our understanding of homomorphisms and their applications in representation theory.
  • Abelian groups provide a clear framework for analyzing homomorphisms because their structure allows easy mapping between different groups while preserving operations. The simplicity of these mappings encourages deeper insights into how different groups relate and interact through their representations. This understanding lays the groundwork for applying more complex concepts in representation theory, highlighting connections between different algebraic structures.

Related terms

Group Theory: A branch of mathematics that studies algebraic structures known as groups, which are sets equipped with an operation that satisfies certain axioms.

Homomorphism: A structure-preserving map between two groups that respects the group operation, allowing for a comparison between different groups.

Characters: Complex-valued functions that provide a way to study representations of groups by encoding information about how group elements act on vector spaces.