Inverse of a coset

Dependencies:

1. Coset

Let $H$ be a subset of group $G$. $H^{-1}$ is defined to be $\{h^{-1}: h \in H\}$.

Then $(gH)^{-1} = H^{-1}g^{-1}$. When $H$ is a group, $(gH)^{-1} = Hg^{-1}$.

Proof

$$ (gH)^{-1} = \{(gh)^{-1}: h \in H\} = \{h^{-1}g^{-1}: h \in H\} = H^{-1}g^{-1} $$

When $H$ is a group, $H^{-1} = H$, because $H^{-1} \subseteq H$ and $|H^{-1}| = |H|$.

Dependency for: None

Info:

• Depth: 2
• Number of transitive dependencies: 2

Transitive dependencies:

1. Group
2. Coset