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:
$\forall\;\;a, b\;\in\; G$,

$\qquad\quad a • b = b • a$

Group

Definition: A group is a set, $G$, together with an operation $•$ that combines any two elements $a$ and b to form another element, denoted by $a • b$ or $ab$.  A group $(G, •)$, must satisfy four requirements known as the group 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 $(G, •)$.