Is every coset of a group closed under taking inverses?

Question

Is every coset of a group closed under taking inverses?

What I mean is that if $G$ is a group and $H$ is a subgroup, and let $a$ be any element of $G$. Then the coset $aH$ is not necessary a group. But does every element of $aH$ have an inverse element also in $aH$?

Answer 1

No, this does not have to hold.

Example: The symmetric group $S_3$ with 6 elements.

$U=\{\operatorname{id}, (12)\}$ is a subgroup.

Now view $(13)U=\{(13), (123)\}$. Then the element $(123)$ has no inverse.

We have $(123)(13)=(23)$ and $(123)(123)=(132)$

Answer 2

No. $a\in aH$. But if $a^{-1}\in aH$, then $a^{-1}=ah\implies h=a^{-2}$. So $a^2\in H$.

So, for instance, consider the dihedral group, $D_4=\langle r,s\mid r^4,s^2, (rs)^2\rangle $.

Take $H\le D_4$ where $H=\{s,e\}$. Then $r^2\not\in H$.