General Linear Group is Group

Theorem
Let $K$ be a field.

Let $\operatorname{GL} \left({n, K}\right)$ be the general linear group of order $n$ over $K$.

Then $\operatorname{GL} \left({n, K}\right)$ is a group.

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