I'm revising questions on groups for exams, and I still can't quite understand what an Abelian group is. Please help me understand, if anyone could give me a more simple explanation.
An Abelian group $G$ is a group $G$ such that the order of multiplication doesn't matter. Precisely: an Abelian group is such that $ab = ba$ for all $a,b \in G$.
An example of an Abelian group: the integers.
A non-example: the group $S_3$ of permutations on 3 letters.
Very simply, Abelian groups are ones which satisfy the additional property of commutativity. That means for all elements $x$ and $y$ in the group $G$, $xy = yx$. So the following are Abelian (or commutative) groups:
The following are groups that are not Abelian: