TITLE: Is $GL(2,\mathbb Z)$ a group?
QUESTION [5 upvotes]: Is the set of $2\times 2$ invertible matrices with integer entries a group under matrix multiplication? I believe not, because inverses for elements in this set may not be in the set (ie, may not have integer entries). Is this correct?

REPLY [4 votes]: For any ring $A$, commutative, with $1$,
$GL(2,A)$ is the set of $2\times 2$ matrices with elements in $A$ with determinant an invertible element of $A$. For $A= \mathbb{Z}$ we get $GL(2,\mathbb{Z})$ the set of $2\times 2$ matrices with integral elements and determinant $\pm 1$. For $A=F$ a field we get us $GL(2,F)$ the $2\times 2$ matrices with elements in $F$ and determinant $\ne 0$. 
So yes,$\ GL(2,A)$ is  a group.