---
tag: 0FCHE902D7
kind: object
name: General linear group
aliases: [gl-n]
msc: [20G15, 15A09]
instance_of: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/General_linear_group
    locator: lead section
same_as:
  wikidata: Q524607
  wikipedia: General linear group
---

## Description

Let $R$ be a commutative ring and let $n \ge 1$ be an integer.
The *general linear group* $GL_n(R)$ is the set of [[invertible|0EB9N0S33C]] $n \times n$ matrices over $R$, with [[matrix multiplication|032Q9FB8MT]] as operation.
It is a [[group|G5S1M9]]: the product of invertible matrices is invertible, the identity element is $I_n$, and the inverse of $A$ is $A^{-1}$.
Equivalently, $GL_n(R)$ is the group of units of the [[matrix ring|0FHVPRAE60]] $M_n(R)$.

## Remarks

For a field $F$, $GL_n(F)$ is isomorphic to the group of linear isomorphisms of $F^n$ to itself.
For $n \ge 2$, the group $GL_n(F)$ is not abelian.
For a finite field with $q$ elements, $|GL_n(\F_q)| = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1})$.
The [[special linear group|0V9PMN0RBH]] is the subgroup of matrices with determinant $1$.

## Explanations

### L1 human

The general linear group collects all the invertible matrices of one size.
We can multiply two of them, and we can undo each of them, so they form a group.
Geometrically, its elements are all the ways to transform space linearly without flattening it.