---
tag: 0V9PMN0RBH
kind: object
name: Special linear group
msc: [15A15, 20G15]
instance_of: [G5S1M9]
uses: [0C3CFHDN6R, 0FCHE902D7]
sources:
  - url: https://en.wikipedia.org/wiki/Special_linear_group
    locator: lead section
same_as:
  wikidata: Q2140900
  wikipedia: Special linear group
---

## Description

Let $F$ be a field and let $n \ge 1$.
The *special linear group* is the set
$$SL_n(F) = \{A \in GL_n(F) : \det A = 1\}$$
of $n \times n$ matrices over $F$ with [[determinant|00KDZRW4C0]] $1$, with matrix multiplication.
It is a subgroup of the [[general linear group|0FCHE902D7]] $GL_n(F)$.

## Remarks

It is the kernel of the group homomorphism $\det \colon GL_n(F) \to F^\times$, so it is a normal subgroup.
The quotient $GL_n(F) / SL_n(F)$ is isomorphic to $F^\times$.
Over $\R$, its elements are the linear maps that keep volume and orientation.
$SL_2(\Z)$ is the analogous group of integer matrices with determinant $1$.

## Explanations

### L1 human

The special linear group collects the matrices with determinant $1$.
In the real case, these maps keep the volume of every region, and they do not flip orientation.
Shears, such as $\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$, are typical elements.