Why does a lattice have to have an inversion center? Indeed all lattices have inversion symmetry, but my teacher said a lattice has to have an inversion center: why? If a lattice doesn't have inversion symmetry, what would happen?
 A: It is necessary to distinguish here between a lattice of points (Bravais lattice) and an actual crystal lattice, which may contain more than one atom in a cell (i.e., where some atoms are not accessible via the elementary translations). The former can be shown to have an inversion center, as discussed in the answer by @Gandalf. The actual crystal however may not have inversion symmetry.
Furthermore, the same crystal lattice may or may not have inversion symmetry, depending on the types of atoms it is filled with: e.g., Diamond and zinc oxide have the same crystal lattice, but the former one has inversion symmetry, while the latter doesn't: in diamond the two sublattices of the diamond lattice are filled with identical itoms, while in zinc oxide one is filled with zinc atoms and the other with the oxygen.
A: In three dimensions a mathematical lattice (which crystallographers call a Bravais lattice) is the set of points $\{m\vec a + n\vec b + p\vec c\}$ where $\vec a, \vec b, \vec c$ are vectors which span space and $m,n,p$ are integers. If we invert a point of the lattice in the point $\vec q$ we get the point $2\vec q - m\vec a - n\vec b - p\vec c$. So if $2\vec q$ is a point of the lattice then inversion in $\vec q$ maps the lattice to itself, so $\vec q$ is an inversion centre for the lattice. In particular, $\vec q = \tfrac 1 2(\vec a + \vec b + \vec c)$ is an inversion centre for the lattice that is not on the lattice itself.
A: Since translation symmetry would cause inversion symmetry, system without inversion symmetry could not have translation symmetry which is indispensable for a lattice.
