What does it mean for a group to be Abelian?

Question

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.

Answer 1

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.

Answer 2

A group is abelian iff its irreducible representation $\rho$ has dimension 1.

Answer 3

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:

1. $\langle \mathbb{Z}, + \rangle$ - The group of integers under addition. For $m + n = n + m$ for all integers $m$ and $n$.
2. $\langle \mathbb{Q} - \lbrace 0 \rbrace, \times \rangle$ - The group of non-zero rationals under multiplicaton. for $xy = yx$ for all rational numbers $x$ and $y$.
3. $\langle \mathbb{R}-\lbrace 0\rbrace, \times \rangle$ - The group of non-zero reals under multiplication.
4. $\langle \mathbb{C} - \lbrace 0 \rbrace, \times \rangle$ - The group of non-zero complex numbers under multiplication.
5. $\langle \mathbb{Z}_n, +_n \rangle$ - The group of integers modulo $n$, under addition modulo $n$.
6. Any group of order at most 4.
7. Any cyclic group (and therefore any groups of prime order, because those are necessarily cyclic).
8. Any group in which every non-identity element is of order 2.

The following are groups that are not Abelian:

1. $GL_n(\mathbb{R})$ - The general linear group of degree $n$ over reals, namely the group of invertible $n \times n$ matrices with real entries, for $n \ge 2$. Matrix multiplication is (in general) not commutative.
2. $S_n$, the symmetric group of degree $n$, for any $n \ge 3$. This is the group of permutations (bijective functions on a set) of $n$ letters under composition. Composition of functions is not commutative in general.
3. $\langle \mathbb{H} - \lbrace 0 \rbrace, \times \rangle$ - The group of non-zero quaternions under (quaternion) multiplication.