Definition: An Abelian group, or a commutative group, is a group which along with the group axioms, satisfies an additional axiom of commutativity, i.e., if (G, •) is an abelian group, then it satisfies the following five axioms:
Closure:
\forall\;\;a, b\;\in\; G, \; a • b \;\in\; G.
Associativity:
\forall\;\;a, b and c\;\in\; G, \; (a • b) • c = a • (b • c).
Existence of identity:
\exists\; an element e\;\in\; G, such that \forall\; a\;\in\; G,
\qquad\quad e • a = a • e = a
Such e is known as the identity element of G w.r.t. •.
Existence of inverse:
For each a\;\in\; G, \exists an element b\;\in\; G, such that,
\qquad\quad a • b = b • a = e
Where e is the identity element, and b is called the inverse of a in (G, •).
Commutativity:
\forall\;\;a, b\;\in\; G,
\qquad\quad a • b = b • a
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