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

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

Sufficient conditions

Arbitrary unions and intersections of symmetric sets are symmetric.

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

Examples

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

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

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

See also

• Absolutely convex set
• Absorbing set – Set that can be "inflated" to reach any point
• Balanced set – Construct in functional analysis
• Bounded set (topological vector space) – Generalization of boundedness
• Convex set – In geometry, set whose intersection with every line is a single line segment
• Minkowski functional – Function made from a set
• Star domain – Property of point sets in Euclidean spaces

References

• R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
• Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi. 
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.