Definition:Inverse of Subset/Group

Definition

Let $\struct {G, \circ}$ be a group.

Let $X \subseteq G$.

Then the inverse of the subset $X$ is defined as:

$X^{-1} = \set {x \in G: x^{-1} \in X}$

or equivalently:

$X^{-1} = \set {x^{-1}: x \in X}$

Examples

Subset of $\R$ under Multiplication

Let $\struct {\R, \times}$ be the multiplicative group of (non-zero) real numbers.

Let $S = \set {-1, 2}$.

Then the inverse $S^{-1}$ of $S$ is:

$S^{-1} = \set {-1, \dfrac 1 2}$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $6$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms