Topological Group Examples Properties Relationship To Other

Almost all objects investigated in analysis are topological groups (usually with some additional structure).

1 Examples

Every group can be trivially made into a topological group by considering it with the discrete topology; such groups are called discrete groups. In this sense, the theory of topological groups subsumes that of ordinary groups.

The real numbers R, together with addition as operation and its ordinary topology, form a topological group. More generally, Euclidean n-space Rⁿ with addition and standard topology is a topological group. More generally still, the additive groups of all topological vector spaces, such as Banach spaces or Hilbert spaceIn mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion, certain linear transformations such as the Fs, are topological groups.

The above examples are all abelianAbstract algebra Algebra Group theory In mathematics, an abelian group is a commutative group, i. a group G ) such that a b b a for all a and b in G''. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions fo. Examples of non-abelian topological groups are given by Lie groupIn mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical anas (topological groups that are also manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefors). For instance, the general linear groupAbstract algebra Algebra Linear algebra Lie groups In mathematics, the general linear group of degree n over a field F (such as R or C , written as GL ''n F , is the group of n ''n invertible matrices with entries from F with the group operation that of o GL(n,R) of all invertible n-by-n matricesAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number with real entries can be viewed as a topological group with the topology defined by viewing GL(n,R) as a subset of Euclidean space R^n×n.

The split-complex numberIn mathematics, the split-complex numbers (also called the Lorentz numbers are an extension of the real numbers defined analogously to the complex numbers. The key difference between the two is that whereas multiplication of complex numbers respects the ss with inverses form a topological group used every day to describe spacetime.

All the examples above are Lie groups (if one views the infinite-dimensional vector spaces as infinite-dimensional "flat" Lie groups). An example of a topological group which is not a Lie group is given by the rational numbers Q with the topology inherited from R. This is a countable space and it does not have the discrete topology. For a nonabelian example, consider the subgroup of rotations of R³ generated by two rotations by irrational multiples of 2π about different axes.

In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication.

2 Properties

The algebraic and topological structures of a topological group interact in non-trivial ways. For example, in any topological group the connected component containing the identity element is a normal subgroup.

The inversion operation on a topological group G gives a homeomorphism from G to itself. Likewise, if a is any element of G, then left or right multiplication by a yields a homeomorphism G → G.

Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps. If G is not abelian, then these two need not coincide. The uniform structures allow to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.

As a uniform space, every topological group is completely regular. It follows that if a topological group is T₀ (i.e. Kolmogorov), then it is already T₂ (i.e. Hausdorff).

The most natural notion of homomorphism between topological groups is that of a continuous group homomorphism. Topological groups, together with continuous group homomorphisms as morphisms, form a category.

Every subgroup of a topological group is itself a topological group when given the subspace topology. If H is a subgroup of G the set of left or right cosets G/H is a topological space when given the quotient topology (the finest topology on G/H which makes the natural projection q : G → G/H continuous). One can show that the quotient map q : G → G/H is always open.

If H is a normal subgroup of G, then the factor group, G/H becomes a topological group, and the isomorphism theorems known from ordinary group theory remain valid in this setting. However, if H is not closed in the topology of G, then G/H won't be T₀ even if G is. It is therefore natural to restrict oneself to the category of T₀ topological groups, and restrict the definition of normal to normal and closed.

If H is a subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup, the closure of H is normal.

3 Relationship to other areas of mathematics

Of particular importance in harmonic analysis are the locally compact topological groups, because they admit a natural notion of measure and integral, given by the Haar measure. In many ways, the locally compact topological groups serve as a generalization of countable groups, while the compact topological groups can be seen as a generalization of finite groups. The theory of group representations is almost identical for finite groups and for compact topological groups.

Topological groups

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1 Examples

2 Properties

3 Relationship to other areas of mathematics

Topics: Topological Group, Examples, Properties, Relationship To Other...