TITLE: Automorphism group of the Leech lattice
QUESTION [5 upvotes]: 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?

REPLY [11 votes]: 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 $-1$.