Automorphism group of the Leech lattice

Question

I have seen that the automorphism group of the Leech lattice is the Conway group $\ Co_0$, which is a finite group.
But for example the lattice $\mathbb Z^n$ has an infinite automorhism group. Can anyone explain me, what is the difference between these two lattices, that results in the (in)finiteness of their respective automorphism groups?

Answer 1

You are confusing isometries of a lattice, and isometries fixing the identity. Each lattice in $\Bbb R^n$, considered as a point set, has infinitely many isometries, since each translation by a lattice vector is an isometry.

What is more interesting is studying the isometries that fix the origin. They form a finite group, always non-trivial, as the map $x\mapsto-x$ is a (central) isometry. The first Conway group is obtained by taking the group of Leech lattice isometries fixing $0$, and factoring out the two-element group generated by $x\mapsto-x$.

In $\Bbb Z^n$ the group of isometries fixing $0$ has $2^nn!$ elements, and are the monomial matrices with nonzero entries either $1$ or $-1$.

Answer 2

I may be wrong, but I suspect that here we consider automorphisms of a lattice as subsets of a Euclidean space, and we thus assume an automorphism to also be an isometry. Unless I'm wrong you were thinking about automorphisms of abelian groups. But, as abelian groups the Leetch lattice and $\Bbb{Z}^{24}$ are isomorphic. They differ only in their metric properties. In my opinion this strongly points at the groups of isometries being relevant here. To get finite groups we also assume that the origin is a fixed point.

In that interpretation an automorphism of $\Bbb{Z}^n$ is fully determined by its action on the short length 1 vectors. Therefore the automorphisms of $\Bbb{Z}^n$ form the group $C_2\wr S_n$ of signed permutations (of coordinates).