Definition:Abelian Group

Definition

Definition 1

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 $G$.

Definition 2

An abelian group is a group $G$ if and only if:

$G = \map Z G$

where $\map Z G$ is the center of $G$.

Additive Notation

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

$x + y$ is used to indicate the result of the operation $+$ on $x$ and $y$.

$e$ or $0$ is used for the identity element. Note that in this context, $0$ is not a zero element.

$-x$ is used for the inverse element.

$n x$ is used to indicate the $n$th power of $x$.

Abelian Group Axioms

Under this regime, the group axioms read:

$(\text G 0)$	$:$	Closure	$\displaystyle \forall x, y \in G:$	$\displaystyle x + y \in G $
$(\text G 1)$	$:$	Associativity	$\displaystyle \forall x, y, z \in G:$	$\displaystyle x + \paren {y + z} = \paren {x + y} + z $
$(\text G 2)$	$:$	Identity	$\displaystyle \exists 0 \in G: \forall x \in G:$	$\displaystyle 0 + x = x = x + 0 $
$(\text G 3)$	$:$	Inverse	$\displaystyle \forall x \in G: \exists \paren {-x}\in G:$	$\displaystyle x + \paren {-x} = 0 = \paren {-x} + x $
$(\text C)$	$:$	Commutativity	$\displaystyle \forall x, y \in G:$	$\displaystyle x + y = y + x $

This notation gains in importance and usefulness when discussing rings.

Also known as

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

The term commutative group can occasionally be seen.

Also see

Results about abelian groups can be found here.

Source of Name

This entry was named for Niels Henrik Abel.

Historical Note

The importance of abelian groups was discovered by Niels Henrik Abel during the course of his investigations into the theory of equations.

Linguistic Note

The pronunciation of abelian in the term abelian group is usually either a-bee-lee-an or a-bell-ee-an, putting the emphasis on the second syllable.

Note that the term abelian has thus phonetically lost the connection to its eponym Abel (correctly pronounced aah-bl).

Sources

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1992: William A. Adkins and Steven H. Weintraub: Algebra: An Approach via Module Theory ... (previous) ... (next): $\S 1.1$: Example $1$

\((\text G 0)\)	$:$	Closure	\(\displaystyle \forall x, y \in G:\)	\(\displaystyle x + y \in G \)
\((\text G 1)\)	$:$	Associativity	\(\displaystyle \forall x, y, z \in G:\)	\(\displaystyle x + \paren {y + z} = \paren {x + y} + z \)
\((\text G 2)\)	$:$	Identity	\(\displaystyle \exists 0 \in G: \forall x \in G:\)	\(\displaystyle 0 + x = x = x + 0 \)
\((\text G 3)\)	$:$	Inverse	\(\displaystyle \forall x \in G: \exists \paren {-x}\in G:\)	\(\displaystyle x + \paren {-x} = 0 = \paren {-x} + x \)
\((\text C)\)	$:$	Commutativity	\(\displaystyle \forall x, y \in G:\)	\(\displaystyle x + y = y + x \)