General Linear Group is Group

Theorem
Let $K$ be a field.

The set of all invertible order-$n$ square matrices over $K$ is a group under (conventional) matrix multiplication.

This group is called the General Linear Group and is denoted $\operatorname{GL} \left({n, K}\right)$, or $\operatorname{GL} \left({n}\right)$ if the field is implicit.

The field itself is usually $\R$, $\Q$ or $\C$, but can be any field.

Proof
Taking the group axioms in turn:

G0: Closure
The matrix product of two $n \times n$ matrices is another $n \times n$ matrix.

The matrix product of two invertible matrices is another invertible matrix.

Thus $\operatorname{GL} \left({n, K}\right)$ is closed.

G1: Associativity
Matrix Multiplication is Associative.

G2: Identity
The Identity Matrix serves as the identity of $\operatorname{GL} \left({n, K}\right)$.

G3: Inverses
From the definition of invertible matrix, the inverse of any invertible matrix $\mathbf A$ is $\mathbf A^{-1}$.

Subgroups of the General Linear Group

 * Special Linear Group
 * Unitary Group
 * Special Unitary Group
 * Orthogonal Group
 * Symplectic Group
 * Triangular Matrix Groups

Related Groups

 * Projective Linear Group
 * Affine Group
 * General Semilinear Group
 * Infinite General Linear Group