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Do you know natural/concrete/appealing examples of right/left cosets in group theory ?

This notion is a powerful tool but also a very abstract one for beginners so this is why I'm looking for friendly examples.

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Oh, I bet there is a Rubik's cube example. Look up cosets Rubik's cube. – PyRulez Mar 21 at 13:32
@PyRulez I just happened to have a Rubik's cube example ready from another question. – pjs36 Mar 21 at 17:16
I wonder whether you prefer cosets of a normal subgroup (as in almost all answers) or, to illustrate the general situation, of a non-normal subgroup, which will inevitably be rather trickier, the easiest case being $D_3$. Of the given answers, Jon’s odd/even number answer is easily the simplest and most familiar to beginners. – PJTraill Mar 22 at 0:04

14 Answers 14

The plane $\mathbb{R}^2$ is a group under addition, and the $x$-axis $\{(a,0)\colon a\in\mathbb{R}\}$ is a subgroup of it. Then the lines parallel to $x$-axis are precisely the cosets of this subgroup.

Instead of $x$-axis, you can take any line through origin; it will be a subgroup, and lines parallel to it will be cosets.

Similarly, $\mathbb{C}^*=\mathbb{C}-\{0\}$ is group under multiplication; think it like a punctured plane. Then $S^1=\{z\in\mathbb{C}\colon |z|=1\}$ is a subgroup, which is a circle with center origin and radius $1$. Its cosets are concentric circles to $S^1$.

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Does it have to be any line through the origin? Wouldn't any line do? – Xenon Mar 21 at 21:52
@Xenon If it doesn't pass through the origin, then it has no identity and isn't closed under addition, so it's not a subgroup. – David Mar 21 at 22:09
@Xenon Yes, David is right, and this illustrates the general fact that only one (and exactly one) of all those cosets is actually a subgroup. Which is the one coset that contains the neutral element (or identity, here in $\mathbb{R}^2$ it is zero, $(0,0)$). – Jeppe Stig Nielsen Mar 23 at 12:00

Probably the example most students will find the most familiar is the set of cosets of the integers modulo some fixed integer.

So for an integer $n$, the cosets of the subgroup $n\mathbb{Z}$ in $\mathbb{Z}$ consists of subset of the form $[a] = \{x\in \mathbb{Z}\mid x\equiv a \pmod n\}$ and if we pick one for each $a$ with $0\leq a\leq n-1$ then we get all the cosets.

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The solutions of a linear system $Ax=b$ form a coset of the null space of $A$.

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Provided there are any solutions at all. – Marc van Leeuwen Mar 23 at 19:40

We've all known about the integers modulo the even integers since we were youngsters: an odd integer plus an even integer, or an even integer plus an odd integer, is odd; the sum of two odd integers or two even integers is even.

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This seems to me to win on simplicity and familiarity, even if some other answers may prove more satisfying to more advanced learners. – PJTraill Mar 21 at 23:52
This is a special case of Tobias' answer. – Kimball Mar 22 at 1:45
@Kimball Indeed, but one that deserves special mention due to how natural it will be to most students. – Tobias Kildetoft Mar 22 at 8:14

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Intgr}{\mathbf{Z}}$If $\Lambda = \Intgr^{2}$ is the integer lattice, $G = \Reals^{2}/\Intgr^{2}$ is the corresponding torus, and $H$ is the image of a line (through the origin of $\Reals^{2}$) having rational slope, then $H$ is a torus knot, and its coets (translates in $G$) fibre $G$.

Cosets of a torus knot

If instead $H$ is the image of a line of irrational slope, i.e., an irrational winding, the complement of $H$ is topologically connected but has uncountably many path components (i.e., comprises an uncountable disjoint union of cosets of $H$). The space of path components, i.e., the space of cosets of $H$ in $G$, has the structure of an unmeasurable set, compare 2000's answer.

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I really like this example for the first time I met it, but it is not so easy for beginners. – projetmbc Mar 22 at 8:42
i wanna eat that donut – MichaelChirico Mar 22 at 23:36
I'm assuming one of the lines on this torus represents $H$ and some other lines represent its cosets, but it's not clear which is which. Or does it matter? Can any of the blue, tan, or purple lines represent $H$ with the other two lines representing cosets of $H$? – Todd Wilcox Mar 23 at 13:43
@ToddWilcox: Qualitatively the cosets are all "translates" (in the torus) of a single curve, and the subgroup is merely the coset containing the identity element. Given the specific parametrization used to generate the plot, the blue curve is the subgroup. (To make the $(12,13)$ structure easier to see, I deliberately didn't plot uniformly-spaced cosets.) – Andrew D. Hwang Mar 23 at 21:57

One example beginners have surely seen is that of (compass) direction. It also showed up when solving trig equations. Namely, the angle $\alpha$, measured in radians, corresponds to the same direction as the angle $\alpha+n\cdot2\pi$. For all integers $n$. Therefore

A direction is a coset of $2\pi\Bbb{Z}$ inside the (additive) group $\Bbb{R}$.

Of course, to be unambiguously specified, a direction needs a point of reference (usually North or the positive $x$-axis), and an orientation (clockwise or counterclockwise).

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Really simple, I like this. – projetmbc Mar 22 at 8:37
Equivalently (but it doesn't involve cosets): a direction is a complex number with modulus $1$. The relationship between these definitions, of course, is the map $\mathbb{R}/2\pi\mathbb{Z} \rightarrow \mathbb{C}$ given by $\theta \mapsto \cos \theta + i \sin \theta$, or equivalently, $\theta \mapsto e^{i\theta}$. – goblin Mar 26 at 12:24

Permutations in $S_n$ are divided into two categories: even and odd. These are the two cosets of the alternating group $A_n$.

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In the dihedral group $G = D_{n}$ the group of symmetries of the regular $n$-gon, which is a group with $2n$ elements, the rotations form a subgroup $H$ of order $n$, and there are two right (or left in this special case as $H \lhd G$) cosets of $H$ in $G$. One of these cosets is $H$ itself, and the other coset consists of the $n$ reflections of $G$.

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The same interpretation extends to symmetries of a circle as the limiting case $n\to\infty$. It's still a subgroup of rotations and a coset of reflections. – Jyrki Lahtonen Mar 29 at 4:10
For sure. That's an infinite dihedral group, though not mentioned in the answer – Geoff Robinson Mar 29 at 7:26

While finite group examples may be easier to first digest, cosets naturally come up in calculus as a way to say what indefinite integrals are: the indefinite integral of an integrable function $f$ is the coset $ \{ F + c : c \in \mathbb R \} = F + \mathbb R$, where $F$ is some antiderivative of $f$.

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Good idea and no so hard to understand. – projetmbc Mar 22 at 8:40

From Wikipedia:
Any coset of a subspace $V$ of a vector space, is an affine space over that subspace.

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And students may have seen this in ODEs - solution spaces to linear ODEs are affine spaces. – Kimball Mar 22 at 1:46

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Intgr}{\mathbf{Z}}\newcommand{\Cpx}{\mathbf{C}}$($G = \Reals^{2}$) Each tile in a tesselation of the Euclidean plane by a lattice $\Lambda$ is a disjoint union of cosets of $\Lambda$ in the additive group $\Reals^{2}$. (In the diagram, green intersections are elements of $\Lambda$, and the purple dots indicate a typical coset. The set of cosets may be viewed as the torus obtained by identifying opposite sides of a tile, and the spiral decorations are collections of cosets.)

A tessellation as a union of cosets

Analogous examples exist in arbitrary dimension. Particularly, the unit circle may be viewed as the space of cosets of $2\pi \Intgr$ in $\Reals$.

In a similar vein, M. C. Escher's Print Gallery, as extended by Lenstra and de Smit may be viewed as a union of cosets of a cyclic subgroup of the multiplicative group $(\Cpx^{\times}, \cdot)$.

(General plane tilings are not unions of cosets, but instead are orbits of a group action. The same is true of spherical and hyperbolic tilings.)

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In real analysis when we want to construct a non-measurable set like Vitali set we use cosets of $\mathbb Q$ in additive group of $\mathbb R$.
Since every cosets of $\mathbb Q$ has non-empty intersection by $[0,1]$, by help of Axiom of choice we can put one point from any cosets of $\mathbb Q$ that these points are in $[0,1]$, and construct a non-measurable set.

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($G = S^{3}$) Hopf fibers are a beautiful example of cosets of a non-normal subgroup, namely a circle subgroup of the multiplicative group $S^{3}$ of unit quaternions. (General coset spaces give additional examples, though perhaps none are as easily visualized and as appealing as the Hopf fibration.)

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This might be a meta-question, but why do you provide several answers in the thread instead of expanding your existing answer with still more examples? – Jeppe Stig Nielsen Mar 23 at 12:07
@JeppeStigNielsen: Briefly, there were existing multiple answers, and it seemed reasonable allow the community to up- or down-vote answers selectively, since they're mathematically distinct (one compact Abelian, one co-compact Abelian (compact quotient), and one non-normal). – Andrew D. Hwang Mar 23 at 22:06

The slide rule is an old analog computing device that can be considered as being based on the quotient group $(\mathbb{R_+^*}, \times)/\{10^k, \ k \in \mathbb{Z} \}$ which could be called as well the "floating point universe". An example:

$$\cdots \ \equiv \ 7530 \ \equiv \ 753 \ \equiv \ 75.3 \ \equiv \ 7.53 \ \equiv \ 0.753 \ \equiv 0.0753 \equiv \cdots $$

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