A double coset is a generalization of a coset in group theory, where instead of considering the left or right multiplication by a single subgroup, we consider both left and right multiplication by two subgroups. This structure is essential in understanding the decomposition of groups and plays a significant role in representation theory and the study of symmetric spaces.

Double Coset

A coset is a subset formed by adding a fixed element to each element of a subgroup within a group, illustrating the partitioning of the group into equal-sized, non-overlapping subsets. Cosets are fundamental in understanding the structure of groups, particularly in the study of quotient groups and the Lagrange's theorem, which relates the order of a subgroup to the order of the entire group.

Coset

Group theory is a branch of abstract algebra that studies the algebraic structures known as groups, which are sets equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. It provides a unifying framework for understanding symmetry in mathematical objects and has applications across various fields including physics, chemistry, and computer science.

Group Theory

Left multiplication

Right multiplication

Single subgroup

Two subgroups

Decomposition of groups

Representation Theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. This approach allows complex algebraic problems to be translated into more manageable linear algebra problems, providing deep insights into the symmetry and structure of mathematical objects.

Representation Theory

Symmetric spaces are a class of homogeneous spaces characterized by their symmetrical properties, where for each point there exists an isometry reversing geodesics through that point. They play a crucial role in differential geometry and representation theory, as they generalize the notion of constant curvature spaces and provide a rich structure for studying Lie groups and algebraic groups.

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