Representation of a Group

Definition

A representation of a group is a way to express group elements as linear transformations or matrices acting on a vector space. This concept allows us to study groups by analyzing their actions on different spaces, making it easier to understand their structure and properties. It connects deeply with the ideas of subgroups and normal subgroups, as well as the way group actions can be seen through cosets and their relationships in larger groups.

5 Must Know Facts For Your Next Test

1. Representations can be finite-dimensional or infinite-dimensional, depending on the vector space used for the representation.
2. The study of representations often involves decomposing them into irreducible components, allowing for a clearer understanding of their structure.
3. Normal subgroups play a crucial role in representations since they relate to the kernels of group homomorphisms, which can influence how representations can be factored.
4. Cosets are essential in understanding how representations can act on quotient groups, providing insight into the structure and classification of these actions.
5. The characters of representations can be used to differentiate between non-isomorphic representations and help in classifying representations according to their properties.

Review Questions

• How does the concept of normal subgroups relate to the representation of a group?
  • Normal subgroups are key when discussing representations because they correspond to kernels of homomorphisms. When a representation is built from a group, the image under this representation must respect the normal subgroup's structure. Thus, normal subgroups can influence how representations can be decomposed or factored through quotient groups, revealing deeper insights into the group's overall behavior.
• In what ways do cosets impact the representation of a group and its actions on vector spaces?
  • Cosets are vital in determining how a group acts on vector spaces through its representations. When considering quotient groups formed by normal subgroups, cosets help illustrate how group elements can create equivalence classes that simplify our understanding of their actions. The relationship between cosets and representations allows us to analyze how group actions partition vector spaces and identify invariant subspaces under those actions.
• Evaluate the significance of irreducible representations in understanding the structure of groups and their associated characters.
  • Irreducible representations are significant because they serve as building blocks for all other representations, much like prime numbers in number theory. By studying these irreducible forms, we gain insight into the group's structure and how its elements behave under various actions. The characters associated with these irreducible representations provide a powerful tool for distinguishing between different types of group actions and understanding their implications for the overall representation theory.

Related terms

Group Homomorphism: A function between two groups that preserves the group operation, mapping group elements in such a way that the image of a product is the product of the images.

Irreducible Representation: A representation that has no non-trivial invariant subspaces, meaning it cannot be decomposed into smaller representations.

Character of a Representation: A function that associates each group element with the trace of its corresponding matrix in the representation, providing insights into the representation's structure.