Systems of coset representatives

Question

I have the following question. Let $H\leq U\leq G$ be (not necessary finite) groups. Let $S$ is a System of Right coset represantatives of $U$ in $G$, i.e. $\bigcup_{s\in R} Us=G$ with $Us\cap Us'=\emptyset$ for all $s,s'\in S$ with $s\neq s'$.

1. Question: Is it possible that there exist Elements $u\in U$ and $s,s'\in R$ with $s\neq s'$, such that

   $$u\cdot sH=s'H$$

2. Question: Same question in the situation, where $S$ is a System of Left coset represantatives.

   In the first situation, I know that such an $u$ can't exist, if $U$ is normal $G$, because $$u\cdot sH=s'H\overset{U\unlhd G}\Leftrightarrow u\cdot H\cdot s=H\cdot s'$$ Since $u\cdot H\cdot s\subseteq U\cdot s$, $H\cdot s\subseteq U\cdot s'$ and $Us\cap Us'=\emptyset$. But what happens when $U$ is not normal?

3. Question: If $S'$ is a System of Left coset represantatives of $H$ in $U$ and $S$ is one of $U$ in $G$. Then it is clear, that $S\cdot S'$ is one of $H$ in $G$. But can I modify $S$ and $S'$, such that $S'\cdot S$ is one of $H$ in $G$???

Thanks for help.

Answer 1

Take $G = S_{3}$, $H = U = \langle (12) \rangle = \{ 1, (12) \}$.

As a system of right coset representatives, take $R = \{ 1, (123), (132) \}$.

Now choose $u = (12) \in U$, and $s = (123) \ne (132) = s'$.

We have $$u s H = (12) (123) H = (13) H = \{ (13), (132) \} = (132) H = s' H.$$

Barring mistakes.