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}


\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!

  • 3
    $\begingroup$ It looks to me like a simple typographical error: $g'g^{-1}\in H$ is clearly correct, and $g'\in H$ is clearly wrong as it would make $[g]=H$. (And your English, while not perfect, is very good indeed.) $\endgroup$ Feb 25 '15 at 10:40
  • $\begingroup$ Thank you for your comment @BrianM.Scott! Then you want to mean that instead of $\{g'\in G \mid (g')^{-1}\in H\}$ it's $\{g' \in G \mid g'g^{-1} \in H\}$? $\endgroup$
    – Relure
    Feb 25 '15 at 11:29
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    $\begingroup$ One point about what you wrote: The collection of right cosets need not form a group, and likewise for the set of left cosets so you shouldn't talk about the quotient group structures, It is precisely when every left coset is a right coset (and conversely) that the collection of (say, right) cosets can be made into a group with opertion inherited from $G$. $\endgroup$ Feb 25 '15 at 11:43
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    $\begingroup$ @Abrahamlure: You're welcome. Yes, that's exactly what I'm suggesting. $\endgroup$ Feb 25 '15 at 13:58

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.


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