Inverse of Subgroup

Theorem

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

Let $H$ be a subgroup of $G$.

Then:

$H^{-1} = H$

where $H^{-1}$ is the inverse of $H$.

Proof

As $H$ is a subgroup of $G$:

$\forall h \in H: h^{-1} \in H$

The result follows.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets: Exercise $9.10 \ \text {(a)}$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $6 \ \text{(ii)}$