Cosets and normal subgroups?

Question

I'm super confused on the following problem:

Let $G$ be a group and let $H$ be a subgroup of $G$. Suppose that the sets $(G/H)_{l}$ and $(G/H)_{r}$ are equal. Does $H$ have to be a normal subgroup of $G$? Justify your answer through a proof or a counterexample.

Answer 1

What this question is asking is to show this:

If the following sets are equal:

$$\{xH \mathrel | x \in G \} = \{Hx \mathrel | x \in G\}$$

Then is $H$ a normal subgroup?

The answer is "yes"!

Details: (from proofwiki)

Say $Hx$ is a right coset of $H$. Then we know there exists a $y \in G$ such that

$$Hx = yH$$

Then we know that $x \in yH$, so $y^{-1}x \in H$.

Then

$$ \begin{split} Hx &= yH \\ &= y (y^{-1} x)H \\ &= xH \end{split} $$

So $H$ is a normal subgroup.

Answer 2

Hint: For any $ a \in G $, $ aH \cap Ha $ is always nonempty for any subgroup $ H $.