"Natural" example of cosets

Question

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.

Answer 1

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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Edit: Consider the group ${\rm GL}_n(k)$ of $n\times n$ invertible matrices over a field $k$ and ${\rm SL}_n(k)$ be the subgroup consisting of matrices with determinant $1$. Then for every $\lambda\in k-\{0\}$, the subset of ${\rm GL}_n(k)$ consisting of matrices with determinant $\lambda$ is a coset of ${\rm SL}_n(k)$ (where $\lambda=1$ gives trivial coset).

Answer 2

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.

Answer 3

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

Answer 4

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.

Answer 5

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Intgr}{\mathbf{Z}}$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 $G$.

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.

Answer 6

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).

Answer 7

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

Answer 8

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 $f$.

Answer 9

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 $G$.

Answer 10

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

Answer 11

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Intgr}{\mathbf{Z}}\newcommand{\Cpx}{\mathbf{C}}$($G = \Reals^{2}$) 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.)

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.)

Answer 12

($G = S^{3}$) 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.)

Answer 13

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.

Answer 14

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:

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