How to visualise operations in quotient subgroup?

Question

I have difficulty in understanding the operations on quotient subgroup. Is there any way to visualise them so that things becomes little clear?

Let $\mathbb{Z}$ be a group with $+$ operation then set oe even number forms a normal subgroup in a $\mathbb{Z}$, so $\mathbb{Z}/2\mathbb{Z}$ is a quotient subgroup. In this quotient subgroup we have two elements.

I know that if I take two elements from the coset like $g_1 + 2\mathbb{Z}$ and $g_2 + 2\mathbb{Z}$ then $(g_1+g_2) + 2\mathbb{Z}$ but I am getting any better way to visualise operations here?

Question : How to visualise operations in quotient subgroup?

Answer 1

It is sometimes nice and/or possible to do it via the Cayley table for the ``mother" group $G$. To do this, you need to order the elements of $G$ along the main header row and column in a particular way.

Suppose you have a finite group $G$ and a normal subgroup $H$. Of course, there are finitely many cosets of $H$ in $G$: maybe call them $H, a_1H, a_2H, \ldots, a_kH$. (Where these are all distinct and exhaust $G$.)

Now write the elements of $G$ in the header row so that they are grouped together by cosets. So, write all the elements of $H$ first (maybe start with $e$), then the elements of $a_1H$ next, and so on. Now use the exact same ordering for the column header.

If you now color all elements of a common coset the same color, using $k+1$ different colors for the $k+1$ different cosets, the Cayley table for $G/H$ can be read off by colors. In other words, the Cayley table for $G$ will come in obvious blocks by color, and these blocks are the elements of $G/H$. An example of $A_4$ modulo a Klein-4 subgroup, stolen from the MAA website*, looks like this:

Exercise: Do this for $G=D_4$, the dihedral group of order $8$, with $H$ the subgroup consisting of the four rotations. The quotient must be isomorphic to $\mathbb{Z}_2$, and you can see this in the Cayley table for $G$ with only two colors.

*Group Visualization with Group Explorer - Conclusion › Author(s): Nathan Carter and Brad Emmons

https://www.maa.org/press/periodicals/loci/joma/group-visualization-with-igroup-exploreri-quotient-groups-in-multiplication-tables

Answer 2

Presumably what the OP means is "how to view operations in quotient groups geometrically" and the best way of doing that in the example $\mathbb Z/p\mathbb Z$ is to view the elements as lying on the unit circle in the complex plane via the representation $e^{\frac{2\pi i}{p}}$. Then the operation is just composition of rotations.

Answer 3

@Randall gave a very good answer! I would also like to give a very rough metaphor that might be helpful to @Complexity when working with quotients. A nice semantic/qualitative way to view the elements of a quotient group $G/H$ is to consider the elements of $H$ as "insignificant" or "inconsequential" (meaning their application has no consequence as they are equivalent to the identity - $0$ or $1$). Then a quotient can be thought of as a thing comprising of group elements that differ an "insignificant" amount from each other ($b-a$ or $a^{-1}b$ being "insignificant" or "inconsequential"). In other words, the "true" elements in this context are the quotients themselves (they are the only things that actually "differ" from each other). Now there are two ways to symbolize them:

1) Symbolize them directly by assigning new names to them like $q_0$, $q_1$, $q_2$, etc.

2) Symbolize them indirectly by using the already available symbols of the original group $G$. To do so, select as representative any (among the "indiscernible") elements of a quotient, namely $g$, and symbolize the quotient as $gH$ (or $g+H$ for abelian groups). This symbolism also reflects the way you can generate the particular quotient as $\{g h, h \in H\}$ (or $\{g + h, h \in H\}$.

Of course, the second indirect way of symbolizing the quotient (or any other equivalence class as a matter of fact) i.e.: via a representative, is an extremely efficient way of doing so and has tremendous cognitive advantages over the first, as it can help the writer directly talk about the properties of the quotient group by relating them to the properties of the underlying group. On the other hand, should one chooses to use the first way, he has to include additional functional symbols for connecting the various $q_i$ to the cosets of $G$ and that would make his expressions quite more cumbersome.

Finally, you can think about the operation in the quotient group the same way you think about the operation in a group as "doing algebra" among its elements (doing "addition" or doing "multiplication" are the common archetypes for abelian and non-abelian groups). It's just that using the second way, you have to remind yourself that you are "talking through representatives", a small cognitive burden in comparison with the extremely expressive power this way has to offer. The only question then is if you are eligible to actually "do algebra" among the cosets and the "normality" of the subgroup $H$ guarantees exactly that!

Another way you can think the handling of the quotients in terms of the second symbolism $gH$ is that you can "pop-in" or "pop-out"/"absorb" elements of $H$ in your expressions anyway you like, for example.

$$gH = (gh)H = (gh^{-1})H$$

Note that the symbolism $G/H$ for a quotient group has also been very cleverly selected since it allows to transfer of some intuition coming from fractions to quotient groups themselves (especially regarding the third isomorphism theorem). For example, we know from athimetic that

$$\frac{g/n}{k/n} = \frac{g}{k}$$

In a symbolically similar way, if $N \subseteq K \subseteq G $ with $N$ and $K$ normal subgroups of $G$ then

$$\left. (G/N) \middle/ (K/N) \right. \simeq G/K $$