General linear group over integers I'm trying to get my head around group theory as I've never studied it before.
As far as the general linear group, I think I've ascertained that it's a group of matrices and so the 4 axioms hold? 
The question I'm trying to figure out is why $(GL_n(\mathbb{Z}),\cdot)$ does not form a group. 
I think I read somewhere that it's because it doesn't have an inverse and I understand why this would not be a group, but I don't understand why it wouldn't have an inverse. 

Integers with multiplication do not form a group. For example the 1x1 matrix (2) has an inverse (1/2) which is not integer.
