Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm. 


I would like to write $\mathbb Z^d = G / H$, where $G$ is the symmetry group of the lattice and $H$ is the stabilizer of a point, but I do not see readily how to do so. This should be an easy question, so maybe one of you can answer it quickly.


Let $V = \{\pm \mathrm e\_1, \cdots, \pm \mathrm e\_d\}$ denote the $2d$ standard basis vectors in $\mathbb Z^d$, and let $H$ be the group consisting of lattice orthogonal matrices from $V$. i.e., an element $\beta \in H$ describes an orthogonal basis of $\mathbb Z^d$, in the sense that it is a matrix whose columns are an independent set from $V$.


The group $G$ should then consist of translations and rotations. Does that mean that it is a semidirect product of $\mathbb Z^d$ and $H$?



The symmetry group of $\mathbb Z^d$
If $G=\mathbb Z^d \rtimes H$ for some group $H$ that acts on $\mathbb Z^d$, there is indeed a natural bijection $G/H \cong \mathbb Z^d$.


As Qiaochu says, what $H$ would be depends on the structure you want to preserve.


The obvious $H$ to choose is $GL\_d(\mathbb Z)$. $\mathbb Z^d \rtimes GL\_d(\mathbb Z)$ is the symmetry group of $\mathbb Z^d$ preserving the affine structure.


Another choice would be $O\_d(\mathbb Z)$ for the standard quadratic form on $\mathbb Z^d$, which would give the group preserving the standard metric structure.


If you want the group to consist of translations and rotations, you need to get rid of the reflections, which you do by setting $H=SO\_d(\mathbb Z)$. This preserves a metric and an orientation.


All the structures are chosen to be translation-invariant so that their symmetry group can act transitively on $\mathbb Z^d$.


