Talk:Abelian group

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Almost a B+ in my opinion... this article could be made real nice real quick. - grubber 15:33, 27 April 2007 (UTC) Exposition is good, but the content doesn't go beyond a few lectures of an introductory abstract algebra course. Needs: more on infinite abelian groups (theory and examples), expansion of relations to other mathematical subjects, main directions of research. The section about automorphisms of finite abelian groups is overly long and should be summarized/moved out. Arcfrk 03:48, 9 July 2007 (UTC)

1 Typo?
2 Capitalization
3 Humor of anonymous edits
4 Translation of Gruppentafel
5 lowercase notation a great honor?
6 Groups from rings
7 List of Small Abelition Groups: Notation
8 Notation for powers
9 Identity
10 Finite abelian groups
11 My favourite mathematical joke
12 Abelian vs. commutative
13 Direct sum/product

[edit] Typo?

Unless I'm too tired, there's an error in the table of small groups. The subgroup list of Z₂⁴ and Z₄² has been interchanged, but I don't know how to correct it...

Now I've changed it. Can someone please check it?

[edit] Capitalization

Wikipedia articles are fairly consistent in writing "Euclidean", "Newtonian", "Eulerian", "Riemannian" with upper-case first letter. Is there any reason why, instead of "Abelian", mostly "abelian" is being used?
S.

We follow the mathematical usage; "abelian" is much more common than "Abelian". Generally, it is considered to be an honour if you have made it to an adjective and are written in lower-case. AxelBoldt

[edit] Humor of anonymous edits

A stalwart from our lexicon of maths jokes to get us through lectures: What's purple and commutes? An abelian grape.

So, in the unlikely event that you were wondering, that's why the recent anon made those pecular edits! Pete/Pcb21 (talk) 12:45, 9 Dec 2003 (UTC)

Could be someone who's central heating's out and has to travel on a very cold bus to work? Dysprosia 12:47, 9 Dec 2003 (UTC)

[edit] Translation of Gruppentafel

I don't know the right word, so I'll post my addition on this talk page, so that someone can put the corrected version in the article.

A finite group can easily be checked to be abelian by creating its group table (what's "Gruppentafel" in english?): The group is abelian iff the table is symmetric along the main diagonal. --SirJective 12:10, 19 Dec 2003 (UTC)

Group table, probably. Would you like me to add this, or do you want to go ahead? I have something to add related to your potention addition... Dysprosia 12:14, 19 Dec 2003 (UTC)

I'm a bit unsure about my language being correct. After 8 years of English lessons at school I can read and understand the most, but never was good at writing things myself. So I would prefer you to add my sentence. It irritates me that I cannot find the group table in this wikipedia, nor via google... A search in MathWorld yields the term multiplication table... That seems to be the right term!

You may also want to look at Cayley diagram or cayley table as it applies specifically to groups.

See also Talk:Cantor-Bernstein-Schroeder_theorem for another addition, which hasn't inspired anyone since the end of october. --SirJective 12:41, 19 Dec 2003 (UTC)

For this, as you can see, I've avoided the problem altogether :) Dysprosia 23:45, 19 Dec 2003 (UTC)

Fine, thanks. 217.80.248.173 12:24, 20 Dec 2003 (UTC)

[edit] lowercase notation a great honor?

Can someone please explain or justify this rather bizarre sounding remark? -Lethe | Talk 15:50, May 5, 2005 (UTC)

First, you are referring to the very first sentence on this talk page. That could have been made explicit, since I first looked at the page history, then at the article, and only later here. But I found it. :)

I think it means to say that once words become generic and widespread, they usually get to be lowercase. Like "gramophone" which I think used to be "Gramophone". Basically the point is that "abelian" is now the defacto spelling, and not "Abelian", which means this lad, called Abel, is very famous now. :) Oleg Alexandrov 16:58, 5 May 2005 (UTC)

So actually I was referring to the final section of the article where it reads:

"The abelian group is rare in being expressed with a lowercase a, rather than A."

I completely missed what that sentence was saying. I thought that somehow denoting abelian groups with a lowercase letter (i.e. let a be the infinite cyclic group) was supposed to be a great honor to the group. Which was fucked up for a whole bunch of reasons: we don't honor mathematical objects, we honor people; most people write groups with capital letters, or at least no one writes the names of ablelian groups in particular as lowercase.

But now I see that the "a" in question was the first letter of the word "abelian", and everything makes perfect sense. Thanks Oleg. -Lethe | Talk 23:02, May 5, 2005 (UTC)

I was thinking about it, and it always seemed to me that when we write abelian group (or sometimes I'll see, for example, euclidean geometry or something), it's lowercase because the word has become so common, not because we're trying to bestow even greater honor on Abel. True, it is a great honor that one of his mathematical inventions is so important and ubiquitous that we deem it a nonproper word. but the lowercase letter isn't there for the honor, it's there to indicate the commonness of the word. Or at least that's my opinion. -Lethe | Talk 22:51, May 12, 2005 (UTC)

[edit] Groups from rings

Every commutative ring gives rise to two abelian groups in the same fashion

Shouldn't it be: "every unital commutative ring"? What if there are no units in a ring? Or is that not possible? Andres 21:45, 8 May 2005 (UTC)

Wikipedia's definition of ring doesn't allow non-unital rings. --Zundark 22:05, 8 May 2005 (UTC)

[edit] List of Small Abelition Groups: Notation

I wanted to have clarification on notation used in the list of small abelian groups. Frequently one encounters, in the subgroup section, things like n x S_n where n is an element of the natural numbers. I have never seen this notation before. For instance, the article states that a subgroup of Klein's 4-group is 3 x Z ₂. What does the 3 represent? Clearly all subgroups of Klein's 4-group contain 2 elements and hence are isomorphic to Z₂. Does the 3 just mean there exist 3 such isomorphic subgroups? If so the fact there are three of them is a bit of a triviality, and I wonder if it only confuses things. -- Shawn M. O'Hare 13:07, 5 November 2005 (UTC)

Yes, that is what is meant. I do not think the number of subgroups of the same type is in general obvious and/or uninteresting.--Patrick 13:52, 5 November 2005 (UTC)

I explained the notation.--Patrick 13:59, 5 November 2005 (UTC)

[edit] Notation for powers

in the table for notation under powers it is unclear what 'na' stands for, is it not-applicable or n * a, given that right underneath we have a^n. i am assuming the former, but i don't know with certainty so do not wish to edit the page. --201.130.133.221

It's just the notation: in additive notation we write nx rather than xⁿ. --Zundark 08:13, 11 April 2006 (UTC)

[edit] Identity

Do ableian groups include the identity element? --HappyCamper 23:39, 14 September 2006 (UTC)

Any group, abelian or not, has an identity element. It's a requirement for it to be a group. - grubber 00:03, 15 September 2006 (UTC)

Hm...I should have known that. Thank you. --HappyCamper 00:28, 15 September 2006 (UTC)

No worries :) Group theory can be a lot to handle some days! grubber 14:35, 15 September 2006 (UTC)

Just to make this clear, because someone could get this wrong: Every group has an identity element ("includes the identity element"). But in an abelian group that is written additively, i.e. with +, the identity element is 0. E.g. every subgroup of the group of integers (under addition) contains the identity element 0, but the even numbers form a subgroup that does not contain 1 (which is not an identity element in the sense of the group we consider). --Hans Adler (talk) 11:36, 28 March 2008 (UTC)

[edit] Finite abelian groups

The large section on finite abelian groups is not really appropriate here. It would be more appropriate in the article on finitely generated abelian groups. But even better, I think, would be to make it a separate article. I intend to do this if there are no objections. --Zundark 09:48, 16 July 2007 (UTC)

[edit] My favourite mathematical joke

Q: What's purple and commutes?

A: An Abelian grape. —Preceding unsigned comment added by 24.15.135.55 (talk) 19:24, 14 October 2007 (UTC)

[edit] Abelian vs. commutative

For groups these two words are clearly synonyms. But when I read "Let G be a commutative group", then a priori I expect it to be written multiplicatively, while when I read "Let G be an abelian group" , I expect it to be written additively. Are there any sources that discuss this question? I couldn't find any, and of course I may just be wrong. --Hans Adler (talk) 11:56, 28 March 2008 (UTC)

[edit] Direct sum/product

I removed direct sum/product from the table comparing multiplicative and additive notation, since I think it is highly misleading to regard the difference between them as merely a difference in notation. Even if $G \oplus H$ and $G \times H$ are isomorphic for abelian groups, the same does not generally hold for sums $\bigoplus G_i$ and products $\prod G_i$ taken over infinitely many groups. —Preceding unsigned comment added by 213.113.150.132 (talk) 21:05, 9 June 2008 (UTC)

Agreed on the removal, for this and other reasons. The table now is much more to the point. For operations other than on group elements, the influence of additive or multiplicative notation is much less standard (where do automorphisms go? if this group acts, how is the action denoted?), and would just pollute the table. JackSchmidt (talk) 21:46, 9 June 2008 (UTC)