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 $H$?

Iwas puzzled about is why $H$ seems smaller than I expected. I think the reason is that you've picked an inner product on $\mathbb Z^n$ (the usual one) and it doesn't have that many symmetries, since any element has only $2n$ nearest neighbours. You would get different answers by choosing a different inner product. – Hugh Thomas Oct 13 '12 at 19:28