Talk:Group theory/Archive 1

Article removed from Good articles
This article was formerly listed as a good article, but was removed from the listing because it is sketchy, not referenced, unbalanced. What is all that eighteenth-century theory of equations doing here? Please note, I am not marking this with any hostile tags; but this is not a good article. Septentrionalis 20:47, 3 March 2006 (UTC)


 * To answer your question about the 18th century theory of equations: the earliest primordial notional of a group was historically that of a group of substitutions on the solutions of a polynomial, through the work of Galois and his predecessors. So it's here because it belongs here, although the history section could be expanded to make this connection a little clearer. Silly rabbit 15:49, 28 May 2007 (UTC)

New To Advanced Math
Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as group theory, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

anyone know about the Abraham-Kurtz-Jamieson relation, if you do please explain it to me.

Simplification
I understand that this is a very complex topic, and I understand that it is very abstract and difficult to explain. But I took lots of math in college, and I still don't understand a single word of this explanation.

Whoever can come up with a reasonable explanation that laypeople can understand will do well. I believe it is possible, with appropriate use of diagrams, analogy and example. (See the Calculus page).

Whoever comes up with a reasonable explanation of group theory and it's applications deserves a Nobel prize! I've been studying group theory for 6 years now and I've never heard a good explanation of it. I can't even name a good book. My sister tells me there is a course on it in the Leaving Cert (the Irish seconary level exam) so I'll have a look at that. There are a few good chemistry books but they concentrate on the applications rather than the maths. Afn

Response
Group theory isn't that hard but it is very easy to make it hard. It is more or less the study of structure. Start real basic and understand the basics, then you'll have a better chance of understanding what comes after that. For example, start with the rigid motions of a square (flips and rotations) and actually make a table as you rotate and flip a square. Cut a square out of paper, lable the corners and trace around it lableing the square you use for reference. Give each position the square can be in a name. Each rotation or flip results in one of the named positions. A flip means flip across a certain diagonal. A rotation is clockwise turning one corner until it is over the next corner. Make a table, like a multiplication table. You get a simple little table and it should make sense to you.

Look at what you can see in your table and why it is true. For example, each flip is it's own inverse because flipping the table over and then back puts you right where you started. Rotations are not their own inverse. You have to rotate a few times before you get back to the beginning. Work out by actually doing all the flips and rotations. The 'operations' are flipping and rotating. They result in certain postions of the square. It identity element is to do nothing. Two operations are inverses if doing one then the other results in getting you back to the 'start' position. You can learn all sorts of things about the structure of that table and it's elements.

Check that this is truly a group. The elements are the actions (flips, rotations, and 'do nothing.') You have to label the corners, and define the elements by the resulting position of the corner. Label the paper trace of the square also because f1 means flip through the original bd diagonal. For example, the identiy is abcd. If you define f1 to be flip it over the diagonal that runs through corners b and d, then the result of f1 is cbad. Do that twice, and you're back to where you started, so f1 is the inverse of f1. For rotations, if m1 is to rotate once clockwise, then if you start in the original position, m1 results in dabc. m2 is to rotate through two corners clockwise. so m2*e is cdab. The elements are the flips and rotations. You can write it all down and verify that this is a group. It is closed. Every element has an inverse and commutes with it. The associative property holds. You can do it and see. It really helps to do this concrete example for yourself.

Now make mulitplication tables for multiplication mod n for some small integers. Look at their structures. Are any the same as the structure of the table for the rigid motion of the square? The same structure means that you can match up the elements so that the identity element matches up for each, and inverses match, etc. You could recode each so that they look the same. They can be mapped onto each other. If the map is perfect, and both structures could essesntially be recoded to look the same then that map is an isomorphism. The operations map to each other, the inverses map to each other, and the identity elements maps to each other.

These are some basic concepts. If you don't understand them you will likely be lost as you do more complicated things. You can think of logrithms as a map of addition to multiplication. 1 gets mapped to 0 (identity of mult to identiy of addition). Adding exponents might be easier than multiplying, so this 'map' makes it easier to do things. You can pop back and forth into whichever structure you want because they are structurally identical.

This is what group theory is good for... When two structures are the same, one might be easier to work in. You can map on over to it, work there, then map your way back.

Sometimes a map takes whole sections of elements and maps them onto a single element of the other structure. When that happens you can't go back and forth because you've lost information. It can still be a handy thing to do. One structure may be easier to understand, and if you can find an easy-to-understand structure to map some difficult structure onto, you can understand the relationships among it's elements because you understand the relationship of the easy structure.

Everything else is just the details. It gets really fun. A pretty good basic book is 'topics in algebra' by i.n. herstein. 71.70.142.189 03:07, 10 March 2006 (UTC)

Followup question
I really appreciated this, thanks. It's a good useful example, and I think it should be put in the main article. But I still haven't had the main question answered - what is a group? I think the simplest explanation would be to say how they are represented on paper. For example:
 * Here is a set, it's an unordered collection: {4, 'a', 0}.

This is not a group. A group is a set together with an operation. Applying the operation to elements in the group always results in another element from the group. There has to be an identity element. Every element has an inverse. For example, the integers mod 5 or (mod any prime) with multiplication is a group. The elements are 1, 2, 3, 4, 0. The operation is mulitiplication mod 5. When you multiply two of these elements, you always get on of these elements for the answer. (the answer is the remainder when you divide the answer by 5; e.g., 4*4 = 16=1mod5, because the remainder is 1 when you divide 16 by 5. A group has to have elements and an operation, and when you perform the operation on 2 numbers (it's always performed on 2 of the elements at a time) the answer is always one of the other elements. There has to be an identity element. Every element has to have an inverse.


 * Integers are used for counting the number of things in sets, here is the number 5 in the simplest representation: *****.
 * Functions are sets of pairs of corresponding inputs and outputs, here is a function that doubles inputs: { 4->8, 5->10, 10->20, ...}

What is a group *made of*? How do you represent one? From the explanation above, it seems a group is just a set plus the operations on that set. Is this correct? YES 81.178.102.64 00:09, 8 March 2007 (UTC)

I wouldn't say, "just." The operation together with the set has to have certain characteristics. There has to be an identity, when you perform the operation you have to get for an answer one of the elements of the set, and every element has an inverse (that results in the identify element.)

The usefulness of groups is in finding two groups that are structurally the same, then mapping them on to each other to make problems easier to solve. For example, log is a map that maps integers with addition onto reals with mult. The identies map on to each other, inverses to inverses, etc. This is the value of groups--finding two groups that are structurally the same, mapping from one to the other to solve problems in the easier group then mapping the answer back.

The definition on this page is good, since I'm no mathematician I'll let someone else copy/paste it: http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Abstract_groups.html

Vote for new external link
Here is my site with group theory example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

http://www.exampleproblems.com/wiki/index.php/Abstract_Algebra#Groups

Examples
(Text in preparation, 26 January 2006) Jon Awbrey 22:06, 26 January 2006 (UTC)

Such as:
 * $$(\mathbb{Z}_{n},\cdot)$$
 * $$(\mathbb{Z}_{n},+)$$
 * Permutations under composition: $$(S_{n},\circ)$$ —Preceding unsigned comment added by NClement (talk • contribs) 01:19, 26 September 2007 (UTC)

equations of degree 5 and above
Although I think it is not of real help, I am suggesting that someone add some material related to why we can't solve 5th degree polynomials analytically. Heard it had something to do with group theory. I guess this forms an application for this theory.

I'm hoping that someone make an attempt to do something like this. Hopefully we'll see this material evolve... --Arun T Jayapal 15:57, 22 March 2006 (UTC)


 * See Quintic equation, all explained there. --Salix alba (talk) 18:27, 22 March 2006 (UTC)

Section: Connection between groups and symmetry
I feel this is the crucial, motivating aspect of groups, the reason that groups are ubiquitous in mathematics (in a way that monoids or Jordan algebras are not).

Every group occurs as a symmetry group. For example, every finite group and every complex Lie group have faithful linear representations. More generally, any group G is the symmetries of the Cayley graph of G with respect to the generating set G: namely, the graph having vertices g &isin; G ; and having directed edges, each colored with an element a &isin; G, of the form (g, ga). (If the group is infinite, this is of course an infinite graph.)

Even more to the point: every important group is important because it is the symmetries of some natural object. Permutations are the symmetries of an unstructured set. Linear groups (even the exceptional simple Lie groups) are symmetries of vector spaces (with various additional structure). The Monster is "explained" by Moonshine.

Peter Magyar

Dept of Math, Michigan State University —The preceding unsigned comment was added by Special:Contributions/ (talk)

I was very glad to see this addition as it was something I was hoping to add myself. However, the additions made seemed to go off topic, were overly advanced, and did not take advantage of wiki links. I removed material which I felt was off topic and advanced. Additionally I tried to make an explicit connection between symmetries and the axioms of a groups. However I feel this can be done more elegantly and has much room for improvement. Padicgroup 18:19, 26 August 2007 (UTC)

One nice comment from the talk page that did not make it to the article was the idea that every group is precisely the group of symmetries of some object. Cayley's theorem just shows a group is a subgroup of the full group of symmetries, but a little more is true. The centralizer of the left regular representation is the right regular representation, which is sort of the right idea (the automorphism group of the G-set G is G), but I like the phrasing using the Cayley graph a little better. Does someone have a reference that states such a thing clearly? JackSchmidt 19:31, 26 August 2007 (UTC)

I do not believe the Cayley graph idea works in general. For instance take a cyclic group. The Cayley is just an n-gon. So the symmetry group of the graph is the dihedral group, not the cyclic group itself. I can not think of any theorem the states any group is the full symmetry group of some object. Cayley's theorem at least says every group is a "partial" symmetry group of some object Padicgroup 19:52, 26 August 2007 (UTC)

The Cayley graph idea does work in general, as long as you think of the Cayley graph as a directed graph. For example, the Cayley graph of a cyclic group is an n-gon with directed edges (i.e. with arrowheads attached to the sides), and the symmetry group of this object is cyclic. In general, you also have to keep track of the labels on the edges (each edge of a Cayley graph is labeled by the corresponding group generator). It is a theorem that G is the complete group of symmetries of its directed, labeled Cayley graph. Jim 20:32, 26 August 2007 (UTC)

I think the Cayley (directed labeled) graph thing should be roughly equivalent to the centralizer of the regular representation theorem which I can find in a lot of places, but I haven't found a textbook statement of the Cayley graph version. I think Cayley graph version would invite a very pretty picture and help encourage the theme of groups acting as (more or less) geometric symmetries, so be superior to what probably looks like an arcane result in permutation groups.

Does anyone have a textbook style reference for the Cayley graph result?

I'll even write the bit in the article if I can have some good model to work from. I find the result counterintuitive and have no feel for geometry, so I definitely need a good source from which to start (to make it intuitive to the readers and give the readers a feel for geometry from a group theory standpoint). JackSchmidt 01:38, 27 August 2007 (UTC)

I don't think one wants Frucht's theorem, but rather Cayley's theorem. The technical difference is Cayley's theorem is about directed colored graphs, and Frucht's theorem(s) are about uncolored graphs or other types of graphs. The important expository difference is Cayley graphs are pretty and Frucht graphs are ugly. Since the point is to show the connection of symmetry and group theory, it seems better to focus on symmetries which are more closely related to the group. JackSchmidt 13:55, 27 August 2007 (UTC)

The example of the sphere is not quite right. The rotations are the symmetries of the sphere with its metric structure AND its orientation (choice of clockwise or counterclockwise on its surface). With LESS structure, the sphere will have MORE symmetries: with only metric structure,  reflections through planes are also symmetries; with only the structure of a manifold, any local stretching (a diffeomorphism) is also a symmetry. Since by definition a symmetry preserves structure, specifying this structure is essential. That is the point I wanted to get across in the progression from a set to a graph to a plane figure. Peter Magyar

I agree on the issue of a sphere so I changed it slightly. However, I think it is important to keep the examples as simple as possible. Group theoryis a top priority article and needs to be understandable to those without a strong mathematical background. As far as the graph theory theorems related to this subject, it is not important to me which is chosen. To me whast is important is that every abstract group is the symmetry group of some concrete object. Others may have other feelings on this. Padicgroup 06:14, 30 August 2007 (UTC)


 * Although I think this is an important and useful topic to discuss, it should probably come after the basic definitions and results, rather than before them: I've moved it further down the page. (I also slightly rejigged the order of other sections for a clearer structure). ArzelaAscoli 13:00, 18 September 2007 (UTC)

Redundancy
It seems to me that there's rather too much stuff in the "Group theory concepts" section, which mostly duplicates the Group article. Unless anyone objects, I'll trim this down in a day or two.

Category Theory
I wish someone would update this page to explain how groups exist in Category_theory. The Groupoid page Groupoid does this in 2 lines but then needs to elaborate with a lot of text to explain why the simple Category Theory definition encompasses so much. —Preceding unsigned comment added by 205.201.57.49 (talk • contribs)


 * A group is a category with a single object in which every arrow is invertible. The first condition makes it a monoid (the multiplication is globally defined), while the second makes it a groupoid (everybody has an inverse).  I think the text you mention could be cut down a bit, or at least expanded from symbols into words.  It should point out that composition of arrows corresponds to multiplication of elements.  Thus the associativity of composition corresponds to the associativity of multiplication, and the existence of identity arrows and inverse arrows corresponds to the existence of identity elements and inverse elements.  The simple category-theoretic definition encompasses so much because categories have internal structure while sets do not, permitting us to avoid repeating things we need for every structure. Michael Slone (talk) 16:47, 3 November 2007 (UTC)

fairly useless? quote
I feel that the quote

"The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing."

showing up in the article is fairly confusing. I'd like to delete it. Anybody against? Jakob.scholbach (talk) 23:32, 23 March 2008 (UTC)