Symmetric set

In mathematics, a nonempty subset of a group 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 $S = S^$ where $S^ := \left\.$
If $G$ is written additively then $S$ is symmetric if and only if $S = - S$ where $- S := \.$

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 $- S \subseteq S.$
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 $S \cup S^$ and $S \cap S^$ are symmetric sets.

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

See also

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References

* R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
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Group theory

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