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.

Definition

In set notation a subset S of a group G is called if whenever s \in S then the inverse of s also belongs to S. So if G is written multiplicatively then S is symmetric if and only if S = S^{-1} where If G is written additively then S is symmetric if and only if S = - S where If S is a subset of a vector space then S is said to be a if it is symmetric with respect to the additive group structure of the vector space; that is, if S = - S, which happens if and only if The of a subset S is the smallest symmetric set containing S, and it is equal to S \cup - S. The largest symmetric set contained in S is S \cap - S.

Sufficient conditions

Arbitrary unions and intersections of symmetric sets are symmetric. Any vector subspace in a vector space is a symmetric set.

Examples

In \R, examples of symmetric sets are intervals of the type (-k, k) with k > 0, and the sets \Z and (-1, 1). If S is any subset of a group, then and are symmetric sets. Any balanced subset of a real or complex vector space is symmetric.