Question

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

Answer 1

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,binG`, we have `a•b in G`

2. `•` is associative.
   For any `a,b,cinG`, we have `(a•b) • (c) = a •(b•c)`

3. `G` contains an identity element
   There exists `einG` such that for all `ainG`, `a•e=e•a=a`

4. Each element of `G` has an inverse in `G`
   For all `ainG` there exists `a^(-1)inG` 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,binG`, we have `a•b = b•a`.

The group `< ZZ, +>` (the integers with standard addition) is an Abelian group, as it fulfills all five of the above conditions.

The group `GL_2(RR)` (the set of invertible `2"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:

`((1,1),(1,0))((1,0),(1,1)) = ((2,1),(1,0))`

but

`((1,0),(1,1))((1,1),(1,0)) = ((1,1),(2,1))`