Group

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

Definition links