User talk:Jim.belk/Dihedral Group Draft

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Contents 1 Initial Comments 2 ELementary properties or examples or dihedral groups as symmetry or... (TBD) 3 Algebraic Properties and so on and so forth 4 Organizing algebraic properties

[edit] Initial Comments

Let me start by saying that - it's great! Please take my comments in that light. And now, my merciless critique :).

Thank you so much for your many helpful comments. It seems best to reply inline. Jim

First, a general philosophical comment. From the point of view of abstract group theory, probably the most "interesting" thing about the dihedral groups is their expositional role as a stepping stone from abelian groups to more complex types of groups. From an applications viewpoint, they're probably most interesting (but not exclusively) as isometries. I'm not sure that both those points of view can be equally represented in one article; the current article is somewhat muddled in this respect. I'd opt for an a more algebraic viewpoint; and so my comments are in that vein. But then, my interests lie primarily in this direction! I would take as an example the page on cyclic group.

I'm a geometric group theorist, so I'd say that my interests are evenly split between the two. Because the geometry is elementary enough to be comprehensible to high-school students and other non-mathematicians, I think it's best if the article starts out with geometry and then moves on to algebra. At the present time, I've mostly written the first part, and I've been having trouble figuring out how to organize the second part. Jim

One other general comment: The "Contents" section is huge. A bit of work in combining similar sections and re-organizing the order of exposition would bring it down to much more manageable level.

You probably noticed later that this was due to the huge fragmentation in the second half of the article. Jim

[edit] ELementary properties or examples or dihedral groups as symmetry or... (TBD)

I'd lose the Cayley diagram. It takes up a big chunk of visual space, and I really find it doesn't further my understanding of what D_3 /is/, or how it works: it's just too many elements to take in visually.

I see what you mean. The goal here is to explain what the elements of D_n are and how they multiply in the most accessible way possible. Looking at it now, I agree that there ought to be a better way of doing this. Jim

On some reflection (heh-heh), there is something interesting to note from looking at the diagram: R*R -> R and S*S -> R; while R*S and S*R both -> S. The graph diagram makes this easy to see, because of the quadrants. So to heighten this, I have colored the quadrants. I'm sure there's a much more elegant way to do this (I'm not much of an HTML hacker); if you know a better way, please go for it! Chas zzz brown

The coloring is great. It highlights exactly what the Cayley table is supposed to show. Jim 00:26, 24 September 2007 (UTC)

The rotating frito chips (very reminiscent of '60s era lineoleum chips!) are good, but something is still lacking - I had to struggle to "see" the reflections (perhaps because of the different colors?). Maybe some sort of A - B - C labeling so we can see the actions "this is the action of reflection 1", "this is the action of reflection 2", "this is the composition of reflection 1 followed by reflection 2, which you can see is a rotation".

I'll work on this. Jim

A suggestion: instead of the reflection axes being labelled 2, 1, 0, label them S_2, S_1, S_0? Or would that confuse the action with that which is being acted upon? Chas zzz brown

I've replaced both of the diagrams in the "group structure" section with SVGs. I changed the labels on the first one to S_0, S_1, and S_2, and I changed the shapes and colors in the second one to make it more clear. Jim 00:26, 24 September 2007 (UTC)

I'd lose the whole "permutations" section. Again, it's a bunch of characters that make sense only because I know that every group action on n elements can be represented as a subgroup of the permutation group on n elements; there's nothing particularly special about D_n in this sense. I'm hardly going to verify that these permutations actually /do/ form D_5; nor does examining these permutations tell me much that isn't already apparent when I consider it as a a reflection group.

Agreed. (Section removed.) Jim

In fact, rather than calling the section "Elementary properties", I'd probably call it "applications" or "examples" of D_n. To me, an elementary property of D_n is that it can be expressed as a semi-direct product; or that it has a single normal subgroup; or that D_n is solvable.

On the other hand, a symmetry composed with a symmetry forms a symmetry; that (and invertability) makes it an /example/ of a group. I wouldn't say that "the group D4 consists of the following eight matrices"; I'd say that that is an /example/ of how D_4 naturally arises in geometry. (I might also note the example of a reflection matrix and a matrix which rotates through an angle not a rational multiple of pi yields a group isomorphic to D_oo).

I changed the wording in the matrix section that you mention. I'm not sure about the change in the title of the first section. What would you think of calling it something like "Symmetries of a polygon"? Jim

It's certainly /the/ classic example; and how it's always introduced. Even my stuffy old copy of "Group Theory", W R Scott refers to it that way ("the isometries of a regular polygon in 2-D Euclidean space form a group, called the dihedral group"). Perhaps, "Definition" as the section title? Then we are defining the elements as the (action of the) reflections and rotations, and defining the group operation as composition of actions. Chas zzz brown

"Definition" is good. Jim 00:26, 24 September 2007 (UTC)

Perhaps also adding another application as noted on the Talk page regarding the checksum algorithm (to show that the applications are not limited to geometry). Also, the dihedral group arises in Galois theory; certain field extensions (and thus polynomials) can be characterized as having Galois group D_n.

I wouldn't go into a detailed description of how the dihedral group is applied in geometry (e.g., the proposed "Symmetry in 3 dimensions"). This article is about the dihedral group and its properties, not about dihedral geometric symmetry. I would defer that discussion to other pages, just as I would defer a discussion about Galois theory to other pages.

I've been consider splitting off some of this material into a separate article on dihedral symmetry (perhaps similar to the article on rotational symmetry). Jim

[edit] Algebraic Properties and so on and so forth

After describing these examples of the dihedral group, I'd collect the various presentations into one section, "Presentations", rather than presentation, alternate presentations, generating sets, other generating sets. For each of these presentations, you could talk about the generators, rather than having separate sections about generators.

Matrix representations are representations; they are not the group itself. No need to place matrices in a section about presentations; with the exception of the examples, they should only otherwise appear in a section about representations (i.e., representation theory).

I'd combine algebraic structure, conjugacy, subgroups, dihedral subgroups, cyclic subgroups, conjugacy and stabilizers, product structure, and "Parity considerations" all into one category: structure. (Personally, I find those little cycle diagrams rather pointless; but people seem to like them!) It might make sense to break this into sub-sections: first, Structure for n odd, then Structure for n even, then (possibly) Structure for D_oo; and discard the "Parity considerations". In the latter approach, I'd place a discussion of "automorphisms" within each section.

I agree with you about the cycle diagrams. What would you think of having Cayley graphs? Jim

I might put a separate section that discusses D_n as having the property of being Coxeter group; e.g., "D_n as a Coxeter group", and identifying the Coxeter element and its number under that section. I'd show that one of the given presentations obeys the definition of a Coxeter group.

The generalization from D_n to D_oo to Dih(G) for arbitrary group G has an instructive algebraic value as well as a geometric interpretation: they're all examples of a certain semi-direct product with C_2; and a certain extension of a given presentation. Again, as an algebraist, I'm not particularly interested in whether or not G is some sort of point symmetry when I consider Dih(G).

Anyway, hope that helps! I'd be happy to help edit the content if you like in this page as an ongoing draft; certainly much of what is here is more coherent and useful than on the current page. Cheers. Chas zzz brown 01:47, 23 September 2007 (UTC)

Wow, thank you so much for your very detailed comments. You are welcome to edit this draft, and I would be particularly grateful for help with the later parts. For reference, I'm going to post a list of the algebraic facts I have gathered, and then we can figure out how to organize them.

I'd like to move material from here to the main article on dihedral groups as soon as it's in acceptable form. As soon as we agree on a title and a replacement for the Cayley table, do you think it would be reasonable to move the first section? Jim 02:54, 23 September 2007 (UTC)

Assuming we rename the "Elemntary properties" to "Definitions" the section looks to be in pretty good shape. ready for further criticism, at any rate! I'll take a look at the algebraic section over the next day or two. Cheers Chas zzz brown 21:29, 23 September 2007 (UTC)

Thank you so much for your help. I've implemented your suggestions and moved the "definition" section to the main article. Jim 01:22, 24 September 2007 (UTC)

[edit] Organizing algebraic properties

I've now organized the second half into a list of the information I've gathered. Jim 19:24, 23 September 2007 (UTC)