# Group

*The article on this topic in the Group Properties Wiki can be found at:* Group

## Contents

## Definition

A **group** is a set equipped with three additional operations:

- A binary operation called multiplication, or product
- A unary operation denoted as called the inverse map
- A constant element denoted

such that the following conditions hold:

## Importance

Groups arise in differential geometry, primarily in the following contexts:

- As symmetries, or automorphisms, of geometric structures
- As structure groups of bundles
- As fundamental groups or higher (co)homotopy and (co)homology groups
- As manifolds themselves. Notions of relevance here are topological group and Lie group