A discrete group is a group that is equipped with the discrete topology, making it a topological group whose underlying topological space consists of isolated points. In this context, every subset of the group is open, and the group operations (multiplication and inversion) are continuous with respect to this topology. The term is used primarily in the fields of abstract algebra, topology, and geometric group theory.
Definition
Let $G$ be a group. Endow $G$ with the discrete topology, i.e., the topology in which every singleton ${g}$ for $g\in G$ is an open set. The pair $(G,\tau_{\text{disc}})$ is then a topological group, and $G$ is called a discrete group.
Key Properties
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Continuity of Operations
- Multiplication $m:G\times G\to G,\ (g,h)\mapsto gh$ and inversion $i:G\to G,\ g\mapsto g^{-1}$ are automatically continuous because the product of discrete spaces is discrete, and any function from a discrete space to any topological space is continuous.
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Closed Subgroups
- Every subgroup of a discrete group is closed (and open) in the discrete topology.
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Compactness
- A discrete group is compact if and only if it is finite. This follows from the fact that an infinite discrete space cannot be covered by finitely many open sets each containing a single point.
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Connectedness
- A discrete group is totally disconnected; its connected components are the singletons.
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Lie Group Structure
- When viewed as a Lie group, a discrete group has dimension zero. Consequently, any Lie group that is also discrete must be a finite group.
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Representation Theory
- The representation theory of discrete groups is often studied via unitary representations on Hilbert spaces, especially in the context of harmonic analysis on locally compact groups. Since a discrete group is locally compact, the Haar measure reduces to the counting measure.
Examples
- Finite groups: Any finite group, such as the cyclic group $C_n$ or the symmetric group $S_n$, becomes a discrete group when equipped with the discrete topology.
- Infinite countable groups: The additive group of integers $\mathbb{Z}$, the free group $F_n$ on $n$ generators, and the rational numbers $(\mathbb{Q},+)$ are all infinite discrete groups.
- Lattices in Lie groups: A lattice $\Gamma$ in a Lie group $G$ (e.g., $\mathrm{SL}_n(\mathbb{Z})$ in $\mathrm{SL}_n(\mathbb{R})$) is a discrete subgroup of $G$.
Applications
- Geometric Group Theory: Discrete groups act by isometries on metric spaces, leading to concepts such as Cayley graphs and word metrics.
- Algebraic Topology: Fundamental groups $\pi_1(X)$ of spaces are often considered as discrete groups, especially when the space $X$ is path‑connected and locally path‑connected.
- Number Theory: Arithmetic groups, including modular groups like $\mathrm{SL}_2(\mathbb{Z})$, are discrete subgroups of Lie groups and play a central role in the theory of automorphic forms.
Related Concepts
- Topological group: A group with a topology that makes the group operations continuous; a discrete group is a special case.
- Locally compact group: Discrete groups are locally compact because every point has a compact neighborhood (namely, the singleton set).
- Totally disconnected group: All discrete groups belong to this class, but not all totally disconnected locally compact groups are discrete (e.g., the $p$-adic integers $\mathbb{Z}_p$).