symmetric set

Definition A subset A of a group G is said to be symmetricPlanetmathPlanetmathPlanetmathPlanetmath if A=A-1, where A-1={a-1:aA}. In other , A is symmetric if a-1A whenever aA.

If A is a subset of a vector spaceMathworldPlanetmath, then A is said to be symmetric if it is symmetric with respect to the additive groupMathworldPlanetmath structureMathworldPlanetmath of the vector space; that is, if A={-a:aA} [1].

0.0.1 Examples

  1. 1.

    In , examples of symmetric sets are intervals of the type (-k,k) with k>0, and the sets and {-1,1}.

  2. 2.

    Any vector subspace in a vector space is a symmetric set.

  3. 3.

    If A is any subset of a group, then AA-1 and AA-1 are symmetric sets.


Title symmetric set
Canonical name SymmetricSet
Date of creation 2013-03-22 13:48:26
Last modified on 2013-03-22 13:48:26
Owner Koro (127)
Last modified by Koro (127)
Numerical id 7
Author Koro (127)
Entry type Definition
Classification msc 20A99
Classification msc 22A05
Classification msc 15-00
Classification msc 46-00