# What is an Abelian group, from a linear/abstract algebra perspective?

Apr 27, 2016

An Abelian group is a group with the additional property of the group operation being commutative.

#### Explanation:

A group  < G , •>  is a set $G$ together with a binary operation •:GxxG->G which fulfill the following conditions:

1. $G$ is closed under •.
For any $a , b \in G$, we have a•b in G

2. • is associative.
For any $a , b , c \in G$, we have (a•b) • (c) = a •(b•c)

3. $G$ contains an identity element
There exists $e \in G$ such that for all $a \in G$, a•e=e•a=a

4. Each element of $G$ has an inverse in $G$
For all $a \in G$ there exists ${a}^{- 1} \in G$ such that a•a^(-1)=a^(-1)•a=e

A group is said to be Abelian if it also has the property that • is commutative, that is, for all $a , b \in G$, we have a•b = b•a.

The group $< \mathbb{Z} , + >$ (the integers with standard addition) is an Abelian group, as it fulfills all five of the above conditions.

The group $G {L}_{2} \left(\mathbb{R}\right)$ (the set of invertible $2 \text{x} 2$ matrices with real elements together with matrix multiplication) is non-Abelian, as while it fulfills the first four conditions, matrix multiplication between invertible matrices is not necessarily commutative. For example:

$\left(\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right) \left(\begin{matrix}1 & 0 \\ 1 & 1\end{matrix}\right) = \left(\begin{matrix}2 & 1 \\ 1 & 0\end{matrix}\right)$

but

$\left(\begin{matrix}1 & 0 \\ 1 & 1\end{matrix}\right) \left(\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right) = \left(\begin{matrix}1 & 1 \\ 2 & 1\end{matrix}\right)$