Symmetric set

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In mathematics, a nonempty subset S of a group G is said to be symmetric if


where S^{-1} = \{ x^{-1} : x \in S \}. In other words, S is symmetric if x^{-1} \in S whenever x \in S.

If S is a subset of a vector space, then S is said to be symmetric if it is symmetric with respect to the additive group structure of the vector space; that is, if S = -S = \{ -x : x \in S \}.


  • In R, examples of symmetric sets are intervals of the type (-k, k) with k > 0, and the sets Z and \{ -1, 1 \}.
  • Any vector subspace in a vector space is a symmetric set.
  • If S is any subset of a group, then SS^{-1} and S^{-1}S are symmetric sets.


  • R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
  • W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.

This article incorporates material from symmetric set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.