Along with its _group operation_, a [group](Groups.md) is defined by its _Group Axioms,_ which are as follows: * [closure under the group operation](Closure%20under%20an%20operation.md) (meaning that third element is still in the same group) ^ea43a4 * associativity ^c44c95 * the existence of an identity element in that group ^a49fb1 * invertibility of the group operation ^1304cc # Abelian group axiom Besides the above [group axioms](Group%20axioms.md) [[Abelian groups]] also require * Commutativity # Group-like structures A mathematical object with one or more of these [axioms](Group%20axioms.md) missing is one of a number of [[Group-like structure]]s. that have been defined. #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory