I was wondering under what conditions cosets are closed under inverses. For normal subgroups, I came up with this result:
Let $G$ be a group and $H$ be a normal subgroup of $G$. Then each coset of $H$ either contains inverses for all elements, or none at all.
More precisely, $a^{-1} \in Hb \iff b^{-1} \in Hb$. Since $b$ can be replaced with any representative of the coset, the existence of an inverse of a general element $a$ is necessary and sufficient for the existence of an inverse of any other element.
Proof: $a \in Hb$ for some $b \in G \iff a = hb$ for some $h \in H$. Then $a^{-1} = (hb)^{-1} = b^{-1}h^{-1}$, so $a^{-1} \in Hb \iff b^{-1}h^{-1} \in Hb = bH$ (since $H$ is normal) $\iff b^{-1 }h^{-1} = bh’$ for some $h’ \in H$ $\iff b^{-1} = bh’h = bh’’$ for some $h’’ \in H$ $\iff b^{-1} \in bH = Hb$.
Is this true? Is it useful? Could someone provide some examples of groups to help verify this, or otherwise provide a counterexample? I’m quite new to group theory.