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Property of group subsets (mathematics) From Wikipedia, the free encyclopedia
In mathematics, a nonempty subset S of a group G is said to be symmetric if it contains the inverses of all of its elements.
In set notation a subset of a group is called symmetric if whenever then the inverse of also belongs to So if is written multiplicatively then is symmetric if and only if where If is written additively then is symmetric if and only if where
If is a subset of a vector space then is said to be a symmetric set if it is symmetric with respect to the additive group structure of the vector space; that is, if which happens if and only if The symmetric hull of a subset is the smallest symmetric set containing and it is equal to The largest symmetric set contained in is
Arbitrary unions and intersections of symmetric sets are symmetric.
Any vector subspace in a vector space is a symmetric set.
In examples of symmetric sets are intervals of the type with and the sets and
If is any subset of a group, then and are symmetric sets.
Any balanced subset of a real or complex vector space is symmetric.
This article incorporates material from symmetric set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.