General Linear Group is Group

Theorem

Let $K$ be a field.

Let $\GL {n, K}$ be the general linear group of order $n$ over $K$.

Then $\GL {n, K}$ is a group.

Proof

Taking the group axioms in turn:

Group Axiom $\text G 0$: 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 $\GL {n, K}$ is closed.

$\Box$

Group Axiom $\text G 1$: Associativity

Matrix Multiplication is Associative.

$\Box$

Group Axiom $\text G 2$: Existence of Identity Element

From Unit Matrix is Unity of Ring of Square Matrices, the unit matrix serves as the identity of $\GL {n, K}$.

$\Box$

Group Axiom $\text G 3$: Existence of Inverse Element

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

$\blacksquare$

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 3$: Examples of Infinite Groups: $\text{(iv) (a)}$
• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $2$. GROUP: Example $6$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(2)$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.7$