The general linear group of order $n$, $GL_n(\mathbb{F})$, over a field $\mathbb{F}$ (usually $\mathbb{R}$ or $\mathbb{C}$) is the group of $n\times n$ invertible matrices over $\mathbb{F}$. The operation is the usual matrix multiplicatoin.
Let $\mathbb{F}$, usually $\mathbb{R}$ or $\mathbb{C}$, be a field. The general linear group of order $n$, $GL_n(\mathbb{F})$ is the set of invertible $n\times n$ matrices over $\mathbb{F}$; if $\mathbb{F}$ is clear, it is occasionally omitted from the notation. Since the determinant is a homomorphism, $GL_n(\mathbb{F})$ is indeed a group; the identity matrix $I_n$ is the identity element. It is not an abelian group for $n\geq 2$ because matrix multiplication does not usually commute.
In the case of a finite field $\mathbb{F}_q$, $|GL_n(\mathbb{F}_q)| = \prod\limits_{k=0}^{n-1}q^n-q^k$; this can be seen by examining the number of linearly independent columns.
$GL_n(\mathbb{F})$ has a multitude of subgroups. Some include those consisting of those matrices with determinant $1$, i.e. the special linear group $SL_n(\mathbb{F})$, invertible diagonal matrices, various orthogonal, unitary, and normal groups, and more.
