# Locally compact group

In

mathematics , a**locally compact group**is atopological group "G" which is locally compact as atopological space . Locally compact groups are important because they have a natural measure called theHaar measure . This allows one to defineintegral s of functions on "G".Many of the results of finite group representation theory are proved by averaging over the group. These proofs can be carried over to locally compact groups by replacement of the average with the Haar integral. The resulting theory is a central part of

harmonic analysis . The theory for locally compactabelian group s is described byPontryagin duality , a generalizedFourier transform .**Examples and counterexamples***Any

compact group is locally compact.

*Anydiscrete group is locally compact. The theory of locally compact groups therefore encompasses the theory of ordinary groups since any group can be given the discrete topology.

*Lie group s, which are locally Euclidean, are all locally compact groups.

*A Hausdorfftopological vector space is locally compact if and only if it isfinite-dimensional .

*The additive group of rational numbers**Q**is not locally compact if given therelative topology as a subset of the real numbers. It is locally compact if given the discrete topology.

*The additive group of "p"-adic numbers**Q**_{"p"}is locally compact for anyprime number "p".**Properties**By homogeneity, local compactness for a topological group need only be checked at the identity. That is, a group "G" is locally compact if and only if the identity element has a compact neighborhood. It follows that there is a

local base of compact neighborhoods at every point.Every closed

subgroup of a locally compact group is locally compact. (The closure condition is necessary as the group of rationals demonstrates.) Conversely, every locally compact subgroup of a Hausdorff group is closed. Every quotient of a locally compact group is locally compact. The product of a family of locally compact groups is locally compact if and only if all but a finite number of factors are actually compact.Topological groups are always

completely regular as topological spaces. Locally compact groups have the stronger property of being normal.Every locally compact group which is

second-countable ismetrizable as a topological group (i.e. can be given a left-invariant metric compatible with the topology) and complete.**References***.

*Wikimedia Foundation.
2010.*

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