Definition:Abelian Group

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Definition

An abelian group is a group $G$ where:

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

That is, every element in $G$ commutes with every other element in $G$.

Equivalently, $G$ is abelian iff $G = Z \left({G}\right)$, where $Z \left({G}\right)$ denotes the center of $G$.

When discussing abelian groups, it is customary to use additive notation, where:

Under this regime, the group axioms read:

$(G0):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \forall x, y \in G$	$:$	$\displaystyle x + y \in G$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Closure Axiom
$(G1):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \forall x, y, z \in G$	$:$	$\displaystyle x + \left({y + z}\right) = \left({x + y}\right) + z$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Associativity Axiom
$(G2):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \exists 0 \in G: \forall x \in G$	$:$	$\displaystyle 0 + x = x = x + 0$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Identity Axiom
$(G3):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \forall x \in G: \exists -x \in G$	$:$	$\displaystyle x + \left({-x}\right) = 0 = \left({-x}\right) + x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Inverse Axiom
$(C):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \forall x, y \in G$	$:$	$\displaystyle x + y = y + x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Commutativity Axiom

This notation gains in importance and usefulness when discussing rings.

The usual way of spelling this is without a capital letter, i.e. abelian, but Abelian is frequently seen.

This entry was named for Niels Henrik Abel.

\((G0):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall x, y \in G\)	\(:\)	\(\displaystyle x + y \in G\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Closure Axiom
\((G1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall x, y, z \in G\)	\(:\)	\(\displaystyle x + \left({y + z}\right) = \left({x + y}\right) + z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity Axiom
\((G2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \exists 0 \in G: \forall x \in G\)	\(:\)	\(\displaystyle 0 + x = x = x + 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Identity Axiom
\((G3):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall x \in G: \exists -x \in G\)	\(:\)	\(\displaystyle x + \left({-x}\right) = 0 = \left({-x}\right) + x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Inverse Axiom
\((C):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall x, y \in G\)	\(:\)	\(\displaystyle x + y = y + x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Commutativity Axiom