A discrete subgroup is a subset of a topological group that is itself a group and is endowed with the discrete topology, meaning its elements are isolated points. This concept is crucial in the study of group actions on manifolds and has significant implications in areas such as geometry, number theory, and the theory of Lie groups.

Discrete Subgroup

discrete subgroup

A topological group is a mathematical structure that combines the properties of both a group and a topological space, where the group operations of multiplication and inversion are continuous with respect to the topology. This dual nature allows for the study of algebraic properties in a topological context, facilitating the analysis of symmetry and continuity in various mathematical and physical systems.

topological group

The discrete topology on a set is the topology where every subset is open, making it the finest topology possible for that set. This topology ensures that all functions from a discrete space to any other topological space are continuous, and it is often used as a basic example in topology due to its simplicity.

discrete topology

isolated points

group actions on manifolds

Geometry is a branch of mathematics concerned with the properties and relationships of points, lines, surfaces, and shapes in space. It encompasses various subfields that explore dimensions, transformations, and theorems to understand and solve spatial problems.

geometry

Number theory is a branch of pure mathematics devoted to the study of the integers and integer-valued functions, exploring properties such as divisibility, prime numbers, and the solutions to equations in integers. It has deep connections with other areas of mathematics and finds applications in cryptography, computer science, and mathematical puzzles.

Your Courses

Relevant Degrees

Your Lessons