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.

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    $\begingroup$ It means $x*y=y*x$ for all $x,y$ in the group. What more do you want to know? $\endgroup$ – Najib Idrissi Jun 2 '14 at 12:26
  • $\begingroup$ Abelian is the same as commutative. $\endgroup$ – lhf Jun 2 '14 at 12:27
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    $\begingroup$ It's alternatively called "commutative" $\endgroup$ – Tom Collinge Jun 2 '14 at 12:27
  • $\begingroup$ thanks that helps, i always see this term but was never told the definition. $\endgroup$ – April Jun 2 '14 at 12:35
  • $\begingroup$ thank you very much David Richbery. $\endgroup$ – April Jun 2 '14 at 15:56

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.

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    $\begingroup$ And I find that "abelian" is spelled in lower case even in wikipedia, although it is named after the famous Norwegian mathematician Niels Henrik Abel, and therefore should be capitalized, to my understanding. Other mathematicians are treated with more respect (e.g. "Gaussian integers", not "gaussian integers", "Lagrangian relaxation", not "lagrangian relaxation"). $\endgroup$ – mathse Jun 2 '14 at 13:07
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    $\begingroup$ Actually, some see it as an even higher honour to have a name associated to an object/definition and not be capitalised, as it means it's truly ingrained in our nomenclature, for instance cartesian plane. $\endgroup$ – Dan Rust Jun 2 '14 at 13:23
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    $\begingroup$ @DanielRust From my experience, people tend to write "Abel" in lowercase simply because they do not know who Abel is or was, while they do know that Gauss or Lagrange are mathematicians from the past. So, if not being known is an honor, then so it be. (Who are these "some", by the way? :-)). "Cartesian plane" has no lowercase hits on google, at least on the first 2 pages ... $\endgroup$ – mathse Jun 2 '14 at 13:48
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    $\begingroup$ @mathse: I absolutely agree with you - people write Abelian in lower case because they think it's a funny latin word, but do not realize it's the name of a person in the first place. Same with Boolean algebra (worse in German, where people write 'boolsch' instead of Boolesch or Bool'sch, as they have no idea who George Boole was) $\endgroup$ – Zane Jun 2 '14 at 16:34
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    $\begingroup$ @FredrikMeyer "I honor Abel every day by working in a building named after him" Honouring him by commuting to work is enough for me. *baddum-tsssh* $\endgroup$ – David Richerby Jun 2 '14 at 18:04

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. Quaternion multiplication is anti-commutative, so $xy = -yx$.

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

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    $\begingroup$ How is that supposed to be helpful to someone who doesn't even know what "abelian" means? $\endgroup$ – Najib Idrissi Jun 2 '14 at 12:49
  • $\begingroup$ Where do you see that in your link? Link doesn't even mention infinite Abelian groups. $\endgroup$ – Zane Jun 2 '14 at 16:52
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    $\begingroup$ Answers are not supposed to be only for person asking a question. Rather, StackExchange is supposed to be a Q&A database/archive for anyone in the future on this topic. Thus, answers that are way over the head of the asker are perfectly acceptable here. I do not understand why people are downvoting this answer. $\endgroup$ – apnorton Jun 2 '14 at 19:33
  • $\begingroup$ Which representation do you refer to when writing "$\rho$"? The link gives a property of (finite) abelian groups, but say that this property characterizes abelian groups. (Does it?) $\endgroup$ – Max Morin Jun 2 '14 at 20:19
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    $\begingroup$ @MaxMorin If you have a counterexample it would be instructive to post it. $\endgroup$ – Alexander Gruber Jun 3 '14 at 1:59

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