---
tag: 75H4HY10Z4
kind: concept
name: Classical group
msc: [20G15, 20H20]
sources:
  - url: https://en.wikipedia.org/wiki/Classical_group
same_as:
  wikidata: Q2285809
  wikipedia: Classical group
---

## Summary

The classical groups are the [[general linear groups|0FCHE902D7]] and [[special linear groups|0V9PMN0RBH]], and the groups of linear maps that preserve a nondegenerate form.
For symmetric, alternating and Hermitian forms these are the [[orthogonal|0TKYAEJ8N7]], symplectic and [[unitary groups|0R9R0Y1HDK]].
The name goes back to Hermann Weyl's book on the classical groups.
Over $\C$ they give all the complex simple Lie groups except five exceptional ones, up to local isomorphism.
Over finite fields, their projective versions give several infinite families of finite simple groups.

## Explanations

### L1 human

A classical group is a group of matrices defined by one natural condition.
The general linear group keeps only invertibility, and the special linear group asks for determinant $1$.
The orthogonal group keeps lengths and angles, as rotations and reflections of space do.
The symplectic group keeps an area-like form, which appears in classical mechanics, and the unitary group keeps the length of complex vectors, as in quantum mechanics.
For example, the rotations of three-dimensional space form the special orthogonal group $\mathrm{SO}(3)$.