Group

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

Definition

A group is a set $G$ equipped with three additional operations:

• A binary operation $*$ called multiplication, or product
• A unary operation denoted as $g\mapsto g^{-1}$ called the inverse map
• A constant element denoted $e$

such that the following conditions hold:

• $a*(b*c)=(a*b)*c\mathrm{\forall }a,b,c\in G$
• $a*e=e*a=a\mathrm{\forall }a\in G$
• $a*a^{-1}=a^{-1}*a=e\mathrm{\forall }a\in G$

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

External links

• Groupprops page
• Wikipedia page