Representation of a group

Definition

A representation of a group is a way of describing the elements of the group as linear transformations on a vector space, often through matrices. This concept connects algebraic structures of groups with geometric and analytic structures, allowing for the study of group actions in a more tangible way. By representing group elements as matrices, one can analyze properties such as symmetry, transformations, and invariance in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. Every finite group can be represented as a permutation group acting on itself, which is a special case of Cayley's Theorem.
2. Representations are classified by their dimensions, with one-dimensional representations corresponding to characters of the group.
3. The study of representations helps in understanding the structure of groups by connecting them to linear algebra and geometry.
4. Irreducible representations are those that cannot be decomposed into smaller representations and are crucial in representation theory.
5. Cayley's Theorem asserts that every group is isomorphic to a subgroup of a symmetric group, showcasing the utility of representation in studying group properties.

Review Questions

• How does Cayley's Theorem relate to the concept of representation of a group?
  • Cayley's Theorem states that every group can be represented as a subgroup of a symmetric group through its action on itself. This means that any abstract group can be seen as a set of permutations, allowing for a concrete matrix representation. This connection between abstract groups and symmetric groups provides a foundation for analyzing group actions and exploring their properties through linear transformations.
• Discuss the significance of irreducible representations in understanding the structure of groups.
  • Irreducible representations play a pivotal role in representation theory because they represent the most basic building blocks from which all other representations can be constructed. By understanding these irreducible forms, one can decompose complex representations into simpler components, which simplifies analysis and computation. This decomposition sheds light on the underlying structure of the group and its symmetries, making it easier to study its properties.
• Evaluate how the concept of representation enhances our understanding of symmetry in mathematical objects.
  • The concept of representation significantly enhances our understanding of symmetry by allowing us to model geometric transformations using algebraic structures. By representing groups through matrices, we can directly analyze how these groups act on vector spaces, making symmetry more tangible. This interplay between algebra and geometry enables mathematicians to apply linear algebra techniques to study symmetries in various contexts, including physics and chemistry, revealing deeper insights into the nature of these mathematical objects.

Related terms

Group Homomorphism: A structure-preserving map between two groups that respects the group operation.

Linear Transformation: A function between vector spaces that preserves vector addition and scalar multiplication.

Matrix Representation: A way to express linear transformations as matrices, allowing for the application of linear algebra techniques.