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I just realized that next Tuesday is Fall break. So we don't have a class then, and that's a pity because a) I won't be able to say the end of the group action-stabilizer-orbit story from today, and b) I will need to tell you via Rhea what concepts/notions are useful on the midterm. So, here it goes: | I just realized that next Tuesday is Fall break. So we don't have a class then, and that's a pity because a) I won't be able to say the end of the group action-stabilizer-orbit story from today, and b) I will need to tell you via Rhea what concepts/notions are useful on the midterm. So, here it goes: | ||
Revision as of 08:03, 2 October 2008
Contents
MA 453 Fall 2008 Professor Walther
I just realized that next Tuesday is Fall break. So we don't have a class then, and that's a pity because a) I won't be able to say the end of the group action-stabilizer-orbit story from today, and b) I will need to tell you via Rhea what concepts/notions are useful on the midterm. So, here it goes:
a) The correct statement of the theorem is:
Suppose G acts on a finite set of points and pick a specific point p. Then the number of elements in G is the product of the number of elements in the stabilizer of p with the number of points in the orbit of p.
In the special case of the 12 centers of the pentagons on a soccer ball there are 12 points in the orbit, and 5 elements that stabilize a given such center (the 5 rotations). So a soccer ball has 60 symmetries.
Note: if we allow soccerball-destroying symmetries we get 120, because then the stabilizer would include the reflections about the chosen point that preserve the pentagon on which it sits.
Now to the MIDTERM, b)
You should know and be able to handle the definitions of group, morphism, subgroup, Abelian, left/right coset, normal subgroup, quotient group G/H, Sym(object), cyclic group, kernel, image, product of groups.
You should know what ZZ/nZZ and U(n) are and be able to calculate with them. In terms of preparation, go through all old HW problems and make sure you know how to do each of them. Midterm questions will be similar.
Here are some basic pointers:
- In order to do any editing, you must be logged in with your Purdue career account.
- If you look under MediaWiki FAQ, you get lots of instructions on how to work with Rhea. Some important things are under item 4 in that manual.
- If you want to do things like $ \sum_{i=1}^\infty 1/i^2 = \frac{\pi^2}{6} $ here then you should look a) at the "view source" button on this page and b) get acquainted with Latex [1], a text-formatting program designed to write math stuff.
Here is some more math, to show yo mathsymbol commands: $ \forall x\in{\mathbb R}, x^2\ge 0 $, $ \exists n\in{\mathbb N}, n^2\le 0 $ where $ {\mathbb N}=0,1,2,\ldots $
If you need to find out a latex command, google for the thing you want to make, latex, and "command". (E.g., google for "integral latex command".)
- If you want to make a new page, all you need to do is to invent one. For example, let's say I want to make a page for further instructions on how to deal with Rhea. I just type "double-left-square-bracket page with more instructions double-right-square-bracket", where of course I use the actual brackets. The effect is: I get a link (initially red) to a page that is the empty set. Once I click it, the link page with more instructions_MA453Fall2008walther turns blue and I am transferred to a newborn page of name as indicated.
Note: it may take a few minutes for the new page to start existing. If you click the red link and nothing happens, wait a bit and try again.
Ideas what to put on Rhea
Course notes, HW discussion, solutions to problems you encountered while using Rhea (how do you upload, make links, post movies, ...)
For week 1, click this link here and on that new page create a page as outlined above. Then move to that page and state your favorite theorem. Why is it you favorite theorem? Have other people he same favorite theorem? Crosslink! Use the math-environment if appropriate.
For week 3: post and discuss the notion/theorem that you have found hardest to understand so far. Alternatively, find somebody else's post and reply to it by explainign how you understand things.
Rhea Questions
Course notes
Some Mathematics Symbols That We Need To Know
Discussion topics
notation/theorems_MA453Fall2008walther
proof helps_MA453Fall2008walther
Homework Discussion
Homework 1, September 4
Homework 2, September 11
Homework 3, September 18
Homework 4, September 25
Homework 5, October 2
Math News
2 new Mersenne primes have been found_MA453Fall2008walther
Math Ed Majors: New strides made for middle schoolers_MA453Fall2008walther
Latex comments
On fonts_MA453Fall2008walther
Reference Pages_MA453Fall2008walther - Websites that provide Latex commands
