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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.

April
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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.

bof
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Fredrik Meyer
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    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"). – mathse Jun 02 '14 at 13:07
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    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. – Dan Rust Jun 02 '14 at 13:23
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    @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 ... – mathse Jun 02 '14 at 13:48
  • @mathse a relevant thread – Dan Rust Jun 02 '14 at 13:55
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    @DanielRust Alright, I see. I would still think that there is a strong(er) statistical relationship between lowercase writing and ignorance than there is between lowercase writing and "ultimate fame". (I'm sure this could be tested.) – mathse Jun 02 '14 at 14:21
  • @mathse I honor Abel every day by working in a building named after him ("Niels Henrik Abels hus"). – Fredrik Meyer Jun 02 '14 at 15:03
  • @FredrikMeyer So why did you spell him in lower case? – mathse Jun 02 '14 at 15:06
  • @mathse It is standard terminology. See for example this mathoverflow post, where they discuss exactly that. – Fredrik Meyer Jun 02 '14 at 15:21
  • @FredrikMeyer From the post, I only see that there is a lot of confusion, rather than a standard. But probably this issue is not worthwhile discussing. – mathse Jun 02 '14 at 15:28
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    @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) – Zane Jun 02 '14 at 16:34
  • @zane Of course, they think that "abelian" is similar as "commutative", "associative", "invertible" or "diagonalizable". "Boole" is also an excellent example - few students nowadays probably know that there has been a person named "George Boole" - but as you see, the community here seems to agree that the opposite is actually true - that these mathematicians are so famous that their names have become integrable parts of mathematical terminology. (Actually, I wonder then why "Euler" and "Gauss" have not yet made it ...) – mathse Jun 02 '14 at 16:44
  • @mathse: agree. Maybe we should add this to the 'relevant thread'? – Zane Jun 02 '14 at 16:48
  • Yes, please. (It is also noteworthy to point out that the opposite may happen as well - I knew a few students who always wrote "Eigenvalue" or "Eigen value" because they thought that there did exist a "Prof. Eigen" ...) – mathse Jun 02 '14 at 16:58
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    @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* – David Richerby Jun 02 '14 at 18:04
  • This comment thread is becoming very off topic. Please move to chat if you plan to discuss this issue further. – Alexander Gruber Jun 03 '14 at 02:01
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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.
M. Vinay
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A group is abelian iff its irreducible representation $\rho$ has dimension 1.

draks ...
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    How is that supposed to be helpful to someone who doesn't even know what "abelian" means? – Najib Idrissi Jun 02 '14 at 12:49
  • Where do you see that in your link? Link doesn't even mention infinite Abelian groups. – Zane Jun 02 '14 at 16:52
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    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. – apnorton Jun 02 '14 at 19:33
  • 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?) – Max Morin Jun 02 '14 at 20:19
  • @max a linear representation. which property? – draks ... Jun 02 '14 at 20:27
  • @zane why should it mention infinite abelian groups? feel free to post your answer... – draks ... Jun 02 '14 at 20:29
  • If you want to use the property “Every irreducible representation of $G$ has dimension $1$” as definition of an abelian group, you better make sure it is equivalent to the definition the rest of the world uses. ;) – Max Morin Jun 02 '14 at 20:48
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    @MaxMorin If you have a counterexample it would be instructive to post it. – Alexander Gruber Jun 03 '14 at 01:59
  • @Zane the link gives necessary and sufficient conditions. Isn't that enough? – draks ... Jun 03 '14 at 05:32
  • @draks... the link gives a necessary and sufficient condition for a representation of a finite abelian group to be irreducible. It does not define an abelian group (and as such does not answer the question). It is similar to answering the question “What is a horse?” with “A horse has four legs.” It is true that they do, but such an answer does not actually answer the question. – Max Morin Jun 03 '14 at 09:02
  • @AlexanderGruber There is none for finite groups. $|G|$ is the sum of squares of irreducible representations, so if they all have dimension $1$ there are $|G|$ of them, meaning $G$ has $|G|$ conjugacy classes. Thus each conjugacy class is a singleton, so each element of $G$ is in the center of $G$. I'd be very surprised if this is true for infinite groups, tough. – Max Morin Jun 03 '14 at 09:05
  • @MaxMorin let's see, maybe someone else knows: http://math.stackexchange.com/q/818986/19341... – draks ... Jun 03 '14 at 09:22