Question about right and left cosets.

Question

I want to do a question about how my algebraic structures professor defined left and right cosets. I'll write here his way to present them.

We first talked about quotient group. Let $G$ be a group, $H\leq G$, and we want to build $G\ /\ H$. We look the particular case of $\mathbb{Z}\ /\ n\mathbb{Z}$ to make the generalization. After a little explanation of the last group, he defined two relations: $\sim$ and $\approx$:

Let $G$ be a group. Let $H\leq G$. Let $g_1$ and $g_2$ $\in G$. We say that \begin{equation} g_1\sim g_2\ \ \text{ if }\ \ g_1\ g_2^{-1}\in H, \end{equation}

and

\begin{equation} g_1\approx g_2\ \ \text{ if }\ \ g_2^{-1}g_1\ \in H. \end{equation}

After that, we proved that they are equivalence relations, and then he defined the quotient groups on whom we were interested:

\begin{equation} G\ / \sim\ = G\ /\ H,\\ G\ / \approx\ = H\ \backslash\ G. \end{equation}

We say that $G\ /\ H$ is the set of the right equivalence classes (I think that in english it's called right coset), and then $H\ \backslash\ G$ is the left coset.

Now it comes the part that I don't understand:

Let $g \in G\ /\ H$. The equivalence class of g is:

\begin{equation} [g]=\{ g'\in\ G \mid g' \sim g \}= \{ g' \in\ G \mid (g')^{-1} \in\ H\}=\{ g' \in\ G \mid g\in Hg \}= Hg \end{equation}

The equivalence classes of the elements of $H\ \backslash\ G$ are similar.

My question is: how he can say that

\begin{equation} \{ g'\in\ G \mid g' \sim g \}= \{ g' \in\ G \mid (g')^{-1} \in\ H\}? \end{equation}

As far as I'm concerned, $\ g'\sim g \implies g'g^{-1} \in\ H$. He can say from this that $(g')^{-1}\! \in H$?

Sorry about this long explanation, but I wanted you to know how my professor deduced these quotient groups, because I haven't seen it in any group theory book. I hope you understood it clearly, despite my english. Thank you!

Answer 1

Brian M. Scott was right, there is a typo mistake on my notes. The class of $g$, or the correct way to arrive to $[g]=Hg$, has to be defined this way:

\begin{equation} [g]=\{ g'\in\ G \mid g' \sim g \}= \{ g' \in\ G \mid g'g^{-1} \in\ H\}=\{ g' \in\ G \mid g\in Hg \}= Hg. \end{equation}

So this completes the definition of the elements of the right coset.