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Mathematics grading:	Start Class	High Importance	Field: Algebra

Contents 1 Article removed from Wikipedia:Good articles 2 New To Advanced Math 3 Simplification 4 Response 5 Followup question 5.1 Vote for new external link 5.2 Examples 5.3 Problems 5.4 equations of degree 5 and above

[edit] Article removed from Wikipedia: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)

[edit] 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.

[edit] 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

[edit] 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)

[edit] 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}.
• 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)

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

[edit] 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

[edit] Examples

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

[edit] Problems

	Hit's a-comin', boys. Tell yore folks hit's a-comin'.
	Thomas Wolfe, O Lost, A Story of the Buried Life

[edit] 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)