General Linear Group to Determinant is Homomorphism/Corollary
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Theorem
Let $\GL {n, \R}$ be the general linear group over the field of real numbers.
Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers.
Let $\det: \GL {n, \R} \to \struct {\R_{\ne 0}, \times}$ be the group homomorphism:
- $\mathbf A \mapsto \map \det {\mathbf A}$
where $\map \det {\mathbf A}$ is the determinant of $\mathbf A$.
The kernel of the $\det$ mapping is the special linear group $\SL {n, \R}$.
Proof
From General Linear Group to Determinant is Homomorphism:
- $\det$ is a group homomorphism.
The special linear group $\SL {n, \R}$ is the subset of $\GL {n, \R}$ such that:
- $\forall \mathbf A \in \SL {n, \R}: \map \det {\mathbf A} = 1$
From Real Multiplication Identity is One:
- $1$ is the identity of the multiplicative group of real numbers.
It follows by definition that $\SL {n, \R}$ is the kernel of the $\det$ mapping.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 47.7$ Homomorphisms and their elementary properties
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Theorem $8.13: \ (3)$