Talk:Group (mathematics)/Archive 5

Problems with symmetry group example, some changes I made

There were a couple of problems with the section "Worked example: a symmetry group" First r₃, was defined as a rotation (clockwiset presumably) of the square 270°, while being shown in the figure as being a left rotation by 90° — these are of course identical rotations — but the descriptions should match to avoid confusion. I chose to make the definition match the figure, since that was easier, but I can see reasons for wanting to go the other way. (I also corrected an error where r₃ was described as rotating left by 270°). Second clockwise rotations were apparently being assumed while counterclockwise rotations were described either as say left rotation by 90° or alternatively as rotating by 90° left. I made the direction of the rotation explicit, and chose the "90° left" formulation for consistency (the other way round would be fine with me also). Paul August ☎ 17:58, 25 August 2008 (UTC)

Thanks for your checking! I have a preference for the 90, 180 and 270 way, just because it is more coherent with r1, r2 and r3, which in turn is somehow showing nicely that we have a cyclic group inside D4. Would you volunteer to a) change the image accordingly and b) cover the labels etc. accordingly? Thanks! This way, we have both sides of the medal (the image and the text) well-polished. Currently, we win only a silver medal (well-polished, though). Jakob.scholbach (talk)18:32, August 25, 2008 (UTC)

Sorry, Jakob but I'm not able to reproduce those images. We can try to find their original author, or ask for help on this. Paul August ☎ 18:43, 25 August 2008 (UTC)

All done, I think. I should mention, the images are just text files. I just flipped the arcs. JackSchmidt (talk) 19:35, 25 August 2008 (UTC)

Thanks Jack. Who knew? Paul August ☎ 20:06, 25 August 2008 (UTC)

It's SVG so its just text markup. And since these were not generate by same insane program (like inkscape) the code actually makes sense. I like the change. (TimothyRias (talk) 09:27, 26 August 2008 (UTC))

Yes I could see that once I'd looked. And after I'd looked I also saw that the SVGs were created by you, nice work! Paul August ☎ 18:48, 26 August 2008 (UTC)

The group operation normally is not a rotation of an object.

The paragraph at the start of the definition and illustration section contains the following sentence:

The abstract symbol "•" is to be understood as a general placeholder for a concretely given operation; for example, it might correspond to normal addition or multiplication, to a permutation of objects, or to a rotation of an object.

The last part is somewhat odd. The group elements may represent a permutation of object or a rotation of an object. The group multiplication in such cases will be composition of such operations. I wanted to change this, but realized talking about composition of operations right in this first paragraph might be a little too much to ask of a beginning reader. Anybody have a better idea for an example of a group operation that can be explained in half a sentence to pretty much anyone? (TimothyRias (talk) 09:34, 26 August 2008 (UTC))

How about " ... or a mapping of an object to itself or to other objects", thus including symmetries and function groups. −Woodstone (talk) 10:26, 26 August 2008 (UTC)

That suffers from the same problem. The symbol "•" does not represent a mapping from an object to itself, in any common example. (although the group elements do.) (TimothyRias (talk) 11:37, 26 August 2008 (UTC))

Yes. How about simply: "The abstract symbol "•" is to be understood as a general placeholder for a concretely given operation; for example, it might correspond to normal addition or multiplication." Jakob.scholbach (talk) 12:25, 26 August 2008 (UTC)

(Indeed, sloppy of me.) It would be a loss not to include function composition, one the most extensive applications of groups. How about: " ... or a applying a mapping to the result of another mapping". −Woodstone (talk) 12:32, 26 August 2008 (UTC)

That would be fine. It would have been nice to have an example of something not so directly related to numbers. But it will definitely serve its main purpose.(TimothyRias (talk) 13:54, 26 August 2008 (UTC))

Well, it is not just any mapping, has to be bijective or so. Actually, this intro is only to tell the reader that the symbol is to be interpreted upon the context. The function example is explained in detail some lines below. This list up there does not have to be exhaustive. I think addition and multiplication will convince everybody that the operation symbol has some abstract meaning. I will write this for now. Jakob.scholbach (talk) 13:24, 26 August 2008 (UTC)

If we're looking for a quick and simple half-sentence placeholder for composition, I would think something like "chaining together of symmetries or permutations"; other possibilities include "sequential application", or just "a combination". As Jakob notes, fuller details are given below, so I think full precision isn't required here; what we do need to do is communicate the general idea (preferably with something beyond arithmetic operations). -- Leland McInnes (talk) 14:26, 26 August 2008 (UTC)

An example beyond basic arithmetic is necessary to avoid being misleading as to how versatile groups can be. How about "addition, multiplication or combining permutations in sequence"? --Tango (talk) 17:25, 26 August 2008 (UTC)

Actually, there was a redundancy I have now removed. It now reads "A group is a set with an operation "•" that combines any two elements a and b to form another element denoted a • b. The abstract symbol "•" is to be understood as a general placeholder for a concretely given operation. To qualify as a group, the set and operation must satisfy four requirements known as the group axioms. Before giving these axioms, the concept of a group is illustrated with two examples: the integers, where the operation "•" corresponds to addition, and the symmetry group of the square, whose group operation "•" is given by concatenating certain geometric transformations.". Fine with everybody? Jakob.scholbach (talk) 18:01, 26 August 2008 (UTC)

Recent reformat of the symmetries of a square images

The table of symmetries of the square was reformatted. The original, which positions the images vertically and to the right, looks much better in my opinion. The current centered version which is horizontal and centered, creates too much white space, especially for large monitors. The edit summary mentioned this was to prevent "akward stacking", but I don't understand what this means. What do others think? Paul August ☎ 19:11, 26 August 2008 (UTC)

Agree preceding vertical stack to the right looked much better. −Woodstone (talk) 20:38, 26 August 2008 (UTC)

Stacking is referred to in one of the manuals of style. It means several images in a row on one side of the page, sometimes so that there is no text beside the images. The MoS suggests alternating sides etc. to avoid creating a "sidebar" effect. WP:BUNCH describes another problem, and maybe more importantly, describes a number of solutions. The MoS sections are probably MOS:IMAGES which references WP:Picture tutorial#Avoiding image "stackups". JackSchmidt (talk) 21:08, 26 August 2008 (UTC)

The symmetries table and the group table did not stack very well (especially as the later scales with the font size while the former doesn't). The previous solution being used was to have the whole float wrapped in one big float, which lead to odd looking white space in the layout. Some of the other more common solutions of the problem, like alternating the floats left and right introduces other problems. In the end, I believe the symmetries table is just to big to have text wrapping around it and guarantee a decent layout on a wide variety of screens. But if all other people disagree, I'd be happy to put it back. (TimothyRias (talk) 08:18, 27 August 2008 (UTC))

Cyclic groups

Under Cyclic groups the article says:

"The eponym for this class of groups is the group of n-th complex roots of unity, given by complex numbers ω satisfying ωⁿ = 1 (and whose operation is multiplication)."

Wouldn't addition of integers in modular arithmetic or the rotational symmetries of a regular polygon be a more familiar illustration of a cyclic group for the general reader ? And I am not at all sure that eponym is being used in the right sense here either. Gandalf61 (talk) 16:06, 1 September 2008 (UTC)

The original cycle considered (I think) were cyclotomic integers, especially the roots of unity (maybe Gauss sum?). I don't think the current sentence makes much sense, but it could be corrected by more clearly indicating the "eponym". Of course, the cyclotomic numbers have not as yet achieved personhood, so the word will probably not be precisely correct, though "historical origin of the name" is a little long-winded. It could also be corrected by not trying to indicate the origin of the word "cyclic", but rather just giving a familiar example like clock arithmetic, or the rotation subgroup of the "worked example". JackSchmidt (talk) 17:23, 1 September 2008 (UTC)

OK, I wrote now that the roots are just "a typical example", which should be fine. I didn't know well the meaning of "eponym". Jakob.scholbach (talk) 18:13, 1 September 2008 (UTC)

Correct wikilink

In the section "Definition and illustration", I've wikilinked the term domain, although I'm not sure I've linked to the correct article. Could someone with more knowledge check this and correct my link if necessary? Thanks SP-KP (talk) 22:01, 4 September 2008 (UTC)

I removed your wikilink. The domain talked about there was more a domain in the sense of a container where we work in. (You may have come across integral domain, which is closer, but has a stronger mathematical meaning, which is why this wikilink is not appropriate here, either). Jakob.scholbach (talk) 18:02, 5 September 2008 (UTC)

Suggest using symbols directly rather than HTML entities

I fixed a bug where a citation contained "z" (generating "z") rather than the desired "§" (generating "§"). Is there some reason this article systematically uses "§" rather than "§"? It'd be easier to read the article while editing, and editing would be less error-prone, if the article simply used "§".

Similarly, I suggest replacing "·" with "·", "→" with "→", "∈" with "∈", "×" with "×", "ρ" with "ρ", and "≠" with "≠". I also suggest replacing "−" with "−", for the same reason that the article uses "–" rather than "–". Eubulides (talk) 18:01, 16 September 2008 (UTC)

I don't know how the "§" comes here. I did not write these, but I do write &minus and → etc for the simple reason that I don't know how to enter them otherwise. Easily, that is (this list is a resort, though) More importantly, I do have trouble distinguishing &minus from the standard hyphen in the editing box, and indeed people even revert such things, probably because they don't know (or don't see the difference).

P.S. No need to be jealous :) Just see the 600 edits that I made. If you are still jealous, simply transform your jealousy into finding a nice topic and getting started! Jakob.scholbach (talk) 18:19, 16 September 2008 (UTC)

• OK, thanks, I left "−" alone but replaced the rest.
• With my browser (Firefox), under the editing window (right under the text that says "Do not copy text from other websites") there is a menu of symbols to insert. For example, you can press on the "§" symbol there, and a "§" will magically pop up whereever you've been editing. Very handy.
• My jealousy was inspired by the current state of the article, which is first-class. I've helped with a few featured articles (I'm most proud of Autism) but Group (mathematics) is better.

Eubulides (talk) 18:44, 16 September 2008 (UTC)

Good. And thanks for your appreciation. My next project will probably be field (mathematics) or vector space (don't know yet which one). Feel free to join in. Jakob.scholbach (talk) 19:52, 16 September 2008 (UTC)

I believe I'm the one who put that entity in the references, for the reason that I have basically blocked from my attention the entirety of the special characters line near the edit box. I'm used to TeX, see, and there we have a name for everything...when it turned out that there was no special name for the § symbol, I looked up its number and used that (Copy/Paste makes it easy). Ryan Reich (talk) 03:01, 20 October 2008 (UTC)

If you like copying and pasting strange characters, you may also find User:KSmrq/Chars useful. —David Eppstein (talk) 03:39, 20 October 2008 (UTC)

Great approach for this article

Great work on this article Jakob. I loved your new approach in writing technical scientific articles. The article seems to progress from a public-friendly (I find the definition and illustration section you're most successful feat being able to motivate a curious average reader) to a technical complicated part suitable for the more advanced mathematics. And I see you're working on Vector space in much the same way. Consider submitting your work as a model for technical Wikipedia articles. Eklipse (talk) 06:23, 19 October 2008 (UTC)

Thank you. Well, if you deem it valuable, why don't you talk to the people over there? Jakob.scholbach (talk) 10:13, 21 October 2008 (UTC)