# Symmetric set

Short description: Property of group subsets (mathematics)

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 $\displaystyle{ S }$ of a group $\displaystyle{ G }$ is called symmetric if whenever $\displaystyle{ s \in S }$ then the inverse of $\displaystyle{ s }$ also belongs to $\displaystyle{ S. }$ So if $\displaystyle{ G }$ is written multiplicatively then $\displaystyle{ S }$ is symmetric if and only if $\displaystyle{ S = S^{-1} }$ where $\displaystyle{ S^{-1} := \left\{ s^{-1} : s \in S \right\}. }$ If $\displaystyle{ G }$ is written additively then $\displaystyle{ S }$ is symmetric if and only if $\displaystyle{ S = - S }$ where $\displaystyle{ - S := \{- s : s \in S\}. }$

If $\displaystyle{ S }$ is a subset of a vector space then $\displaystyle{ S }$ 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 $\displaystyle{ S = - S, }$ which happens if and only if $\displaystyle{ - S \subseteq S. }$ The symmetric hull of a subset $\displaystyle{ S }$ is the smallest symmetric set containing $\displaystyle{ S, }$ and it is equal to $\displaystyle{ S \cup - S. }$ The largest symmetric set contained in $\displaystyle{ S }$ is $\displaystyle{ S \cap - S. }$

## Sufficient conditions

Arbitrary unions and intersections of symmetric sets are symmetric.

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

## Examples

In $\displaystyle{ \R, }$ examples of symmetric sets are intervals of the type $\displaystyle{ (-k, k) }$ with $\displaystyle{ k \gt 0, }$ and the sets $\displaystyle{ \Z }$ and $\displaystyle{ (-1, 1). }$

If $\displaystyle{ S }$ is any subset of a group, then $\displaystyle{ S \cup S^{-1} }$ and $\displaystyle{ S \cap S^{-1} }$ are symmetric sets.

Any balanced subset of a real or complex vector space is symmetric.