Definition:General Linear Group
Jump to navigation
Jump to search
Definition
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 (of degree $n$) and is denoted $\GL {n, K}$, or $\GL n$ if the field is implicit.
The field itself is usually $\R$, $\Q$ or $\C$, but can be any field.
General Linear Group over Vector Space
Let $V$ be a vector space.
The group $\GL V$ is the group of all invertible linear transformations of $V$.
Also denoted as
Some sources use the notation $\map {\operatorname {GL}_n} K$ instead of $\GL {n, K}$.
If $K$ is a Galois field of order $q$, the notations $\map {\operatorname {GL}_n} q$ and $\GL {n, q}$ are also seen.
Some sources use $\map {\operatorname {Gl} } {n, r}$ for $\GL {n, \R}$.
Also see
- Results about the general linear group can be found here.
Subgroups of the General Linear Group
- Definition:Special Linear Group
- Definition:Unitary Group
- Definition:Special Unitary Group
- Definition:Orthogonal Group
- Definition:Symplectic Group
- Definition:Triangular Matrix Group
Related Groups
- Definition:Projective Linear Group
- Definition:Affine Group
- Definition:General Semilinear Group
- Definition:Infinite General Linear Group
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: $(2)$
- 1992: William A. Adkins and Steven H. Weintraub: Algebra: An Approach via Module Theory ... (previous) ... (next): $\S 1.1$ Example $7$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.7$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): general linear group
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): general linear group