Definition:Abelian Group
Contents
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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Specifically: Check Definition:Additive Group of Integers in the below
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- 1992: William A. Adkins and Steven H. Weintraub: Algebra: An Approach via Module Theory ... (previous) ... (next): $\S 1.1$: Example $1$
