Due March 30, 2016

• Which topics and theorems do you think are the most important out of those we have studied?

I think the most important thing, besides the theorems, is knowing examples of all different types of groups, which is what I feel the least comfortable on.

• What kinds of questions do you expect to see on the exam?

Just like the other tests, I expect to see opportunities to complete definitions, provide examples, and prove a few things.

• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Wednesday.

**INDEX OF GROUPS, left and right cosets, order of a group and an element, lots of examples of things on the review sheet, 

7.8, Due March 28th

1)  The hardest thing for me to understand in this section are simple groups and composition factors - hopefully we can spend a little bit more time on those in class.  I actually don't even understand the notation on page 226 in the second edition.
2)  I thought it was interesting to see how much simpler it is to prove the first isomorphism theorem for groups than it is for rings, but it makes sense.

7.7, March 25th 2016

1) The hardest thing for me in this chapter are the examples, like on page 218, both of the examples, especially the one where it is showing N+(a,b) some ordered pair.  I also didn't really follow the proof for theorem 7.38.
2) It is interesting to see how quotient groups compare and contrast to quotient rings.

Rest of 7.6, Due March 23rd

1) The hardest part for me to understand was the proof of theorem 7.34.  But even then, it's not too bad.  I think it will be made much more clear in class.
2) I am hoping that in class we can see a lot more examples of this section.  And also, I am just wondering how this has application in the real world, or if it even does.  Sometimes in this class it just seems like a bunch of mathematicians got together and decided on a bunch of definitions of "abstract" objects and then played around with them based on the definitions and axioms that were pre-determined.  Is that true?  It seems somewhat like a game.

7.5, 7.6, Due March 21

1)  We sort of rushed through Lagrange's Theorem last class period, so I hope we talk about it more today and how we can use it in application.  What is the difference between a normal subgroup and a commutative subgroup.  Is there a difference?
2)  I think it is cool that we are still trying to create a list of every finite group.  I am reading the second edition - is there a lot more progress on this problem since this book was published?

7.5, Due March 18th 2016

1) At first, it was super weird for me to think that it would be ab^(-1) is in K, but now it make sense because it totally depends on the operation that we have assigned to our group.  I don't totally understand the proof for Lagrange's theorem - hopefully it will make sense in class.
2)Everything is pretty straightforward.  I think the most interesting theorem is Lagrange's theorem, although I do not yet understand the proof.

7.9, Due March 16th

1) So if I understand correctly, elements that map to themselves to not add to the length k of the cycle.  Can you compose any two groups, even if they have different cycle lengths?  Does it all depend on the amount of elements that they have?  I really don't understand the factorization of permutations.  Both theorem 7.51 and the proof is a little bit difficult.
2)  Even though I don't understand it yet, I would say the most interesting part is seeing how they factor permutations.

7.4, due pi day

1) I guess I don't why we look at an automorphism in groups but not rings.  Is it because an automorphism in rings is just trivial?  I also don't totally understand Cayley's Theorem.  What does it mean for a group G to be isomorphic to a group of permutations?  I don't see how that can be true if the cardinalities are different?
2)  Other than that everything is pretty straightforward - it is cool to see the parallels and differences between groups and rings, and it is easier to understand the proofs and ideas since we started with rings.  I wonder how it would be different if we did groups before rings.

7.3, due March 11

1)  Is the center of a group just the commutative part of the group?  When referring to subsets, subrings, or subgroups, what does proper mean?  That it isn't the whole set.
2)  This section was pretty straightforward.  I am just trying to figure out the application of groups and why we have to define things like this?

7.2, Due March 9, 2016

1)  It says that some exponent rules hold, and others do not, for example (ab)^n may not equal a^n*b^n.  But then we have theorem 7.7 which says that a^m*a^n=a^(m+n), and (a^m)^n=a^(mn).  So is it safe to say that exponent rules don't really hold when the bases are different, but they do when the bases are the same?  And also, the operation in the group could be addition, so when you say a^3, it really just means add a to itself 3 times, right?  I don't really understand theorem 7.8, and corollary 7.9.
2) I think groups are really cool because they have all of these properties that I didn't consider and they are a little bit more abstract and challenging for my mind to wrap around than rings.  It is a good stretch.

7.1, Due March 7th 2016

1) In theorem 7.2, why didn't they just say R is a group under multiplication?  Why did they create an equivalent set called U?
2) It's cool that every ring is an abelian group under addition.  Until I read the proof, I didn't fully believe it.

7.1 (to page 164), Due March 4, 2016

1)  I have heard of permutations before.  Is there any way you could go through some examples of applications of permutations?  As far as I understand, it is just a bijective function, but does it only apply to groups instead of to rings (which we are used to working with)?  With groups we just have stars because it isn't necessarily adding and multiplying like we are used to?  Are we just defining an operation?
I know that order can refer to the power in a polynomial.  Is cardinality of a set just a different meaning of the word order, or are they related in some way?
I'm not sure I totally understand what a symmetric group is.

2)  Could we consider some of the ciphers that we did in cryptography as permutations?

Due March 2, 2016

• Which topics and theorems do you think are the most important out of those we have studied?
• first isomorphism theorem
• rings/homomorphisms,isomorphisms,fields, etc.
• ideals

• What kinds of questions do you expect to see on the exam?
• Lots of definitions that we need to give (similar to the last exam), problems like the homework but not crazy hard, and proofs.

• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Wednesday.
• What is kernal again, how can we find it and why is it important?
• For an integral domain, why do we need to specify that 1 does not equal 0.  Same thing with field?
• For number 3 on the practice problems, how do we do injectivity?
• How can we know for sure that we have an irreducible polynomial?  What are the most important things to know about polynomial rings?
• HOW DO YOU PROVE AN IDEAL IS MAXIMAL?!
• Can we go over examples on the study guide?