Symmetric set

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

is written multiplicatively then

is written additively then

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

The symmetric hull of a subset

is the smallest symmetric set containing

The largest symmetric set contained in

Arbitrary unions and intersections of symmetric sets are symmetric.

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

examples of symmetric sets are intervals of the type

is any subset of a group, then

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.

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