General Linear Group is Group
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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 $1.7$