Processing math: 0%

Abelian Group

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

No comments:

Post a Comment