"Natural" example of cosets 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. 
 A: From Wikipedia:
Any coset of a subspace $V$ of a vector space, is an affine space over that subspace.
A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Intgr}{\mathbf{Z}}\newcommand{\Cpx}{\mathbf{C}}$($G = \Reals^{2}$) 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.)
A: 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.
A: 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 $S^1$.

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).
A: ($G = S^{3}$) 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.)
A: 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.
A: The solutions of a linear system $Ax=b$ form a coset of the null space of $A$.
A: 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.
A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Intgr}{\mathbf{Z}}$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.
A: 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).
A: 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:
$$\cdots \ \equiv \ 7530 \ \equiv \ 753 \ \equiv \ 75.3 \ \equiv  \ 7.53 \ \equiv \ 0.753 \ \equiv  0.0753 \equiv \cdots $$
A: Permutations in $S_n$ are divided into two categories: even and odd. These are the two cosets of the alternating group $A_n$.
A: 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 $f$.
A: 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 $G$.
