1st question
Trying to choose which topics/theorems are important? I would assume all of them, but mostly rings, groups, ideals, quotient rings, quotient groups. Also, Lagrange's Theorem, Cauchy's Theorem, Division algorithm, polynomials, congruence classes.
2nd question
I need to understand the finite abelian groups and sylow theorems as well as direct products...basically chapter 8 is scary to me. A mathematical question: how do you compose k-cycles? I'm sure I did those wrong.
3rd question
The things I learned will most likely not be used directly after this semester. However, like any math course, this class has helped me become better at proving and thinking. Also, it has been interesting. So I will take from this course a greater appreciation for algebra and a sharpened intellect and ability to prove things--which I value very highly. It's been a good course, though tough at times.
Tuesday, April 12, 2011
Sunday, April 10, 2011
Section 8.3, due April 12
What was most difficult?
The most difficult thing is keeping the definitions straight, because there are definitions embedded within definitions. Let me give an example: A sylow p-subgroup is a subgroup of order p^n when p^n is the largest power of p that divides |G|. So, to know what a slow p-subgoup is, you have to wrap your head (in my case, not a smart head) around this definition. Now, In Corollary 8.16, we reference the name sylow p-subgroup, but then I have to embed that definition in 8.16 because I still havnen't wrapped my head around what a sylow p-subgroup is. This is a common thing in mathematics, but when all the terms are new to you, it gets confusing. I have to keep referring back to the previous page. It is a whimpy complaint, I know.
Reflections
Not much to reflect about on this, but I was at the MAA conference on Friday, and a fellow BYU student, Kyle Stewart, gave a presentation in representation theory. He used lots of the vocabulary we have covered in this class. Even though I couldn't follow everything he was saying, I was able to a little bit because of this course.
That was fun!
The most difficult thing is keeping the definitions straight, because there are definitions embedded within definitions. Let me give an example: A sylow p-subgroup is a subgroup of order p^n when p^n is the largest power of p that divides |G|. So, to know what a slow p-subgoup is, you have to wrap your head (in my case, not a smart head) around this definition. Now, In Corollary 8.16, we reference the name sylow p-subgroup, but then I have to embed that definition in 8.16 because I still havnen't wrapped my head around what a sylow p-subgroup is. This is a common thing in mathematics, but when all the terms are new to you, it gets confusing. I have to keep referring back to the previous page. It is a whimpy complaint, I know.
Reflections
Not much to reflect about on this, but I was at the MAA conference on Friday, and a fellow BYU student, Kyle Stewart, gave a presentation in representation theory. He used lots of the vocabulary we have covered in this class. Even though I couldn't follow everything he was saying, I was able to a little bit because of this course.
That was fun!
Thursday, April 7, 2011
Section 8.2, due April 8
What is the most difficult part?
The theorems are straightforward, the proofs are the same (but a bit boring). The difficult things is finding relevance right now. I feel like we've beat groups to death, which I know isn't the case. I'm sure there's MUCH much more out there.
Reflection
I've been way busy, getting ready for the MAA conference (at which I'm presenting) tomorrow. My computer crashed, and being an idiot, I didn't have back up, so I've spent the last two days working on that. Tomorrow, I'll be there, so I won't be in class (again). I feel bad and disrespectful to you for missing class, but I definitely haven't been sitting at home. 17 credits and trying to graduate...yuck. Things got crazy at the end.
The theorems are straightforward, the proofs are the same (but a bit boring). The difficult things is finding relevance right now. I feel like we've beat groups to death, which I know isn't the case. I'm sure there's MUCH much more out there.
Reflection
I've been way busy, getting ready for the MAA conference (at which I'm presenting) tomorrow. My computer crashed, and being an idiot, I didn't have back up, so I've spent the last two days working on that. Tomorrow, I'll be there, so I won't be in class (again). I feel bad and disrespectful to you for missing class, but I definitely haven't been sitting at home. 17 credits and trying to graduate...yuck. Things got crazy at the end.
Tuesday, April 5, 2011
Section 8.1, due April 6
What was most difficult?
After reading this section, I asked myself, "what was that all about?", which is not a good sign!
let me try and summarize what I DO understand:
the direct product is the cartesian product of several groups, and is itself a group.
but a particular group is not a subgroup of the direct product.
Normal subgroups whose intersection is the cyclic group of the identity have abelian elements, where an element is from each group.
it's a bit clearer what I don't understand: the 2nd example on pg. 245, specifically when it moves to page 246, theorem 8.1, and theorem 8.3
Reflections
Just trying to stay alive with all my classes, and MAA conference (I'm presenting!) on Friday.
After reading this section, I asked myself, "what was that all about?", which is not a good sign!
let me try and summarize what I DO understand:
the direct product is the cartesian product of several groups, and is itself a group.
but a particular group is not a subgroup of the direct product.
Normal subgroups whose intersection is the cyclic group of the identity have abelian elements, where an element is from each group.
it's a bit clearer what I don't understand: the 2nd example on pg. 245, specifically when it moves to page 246, theorem 8.1, and theorem 8.3
Reflections
Just trying to stay alive with all my classes, and MAA conference (I'm presenting!) on Friday.
Sunday, April 3, 2011
Section 7.10, due april 4
What was the most difficult part?
The most difficult part was the proof of theorem 7.52. It was long and I found it difficult to pay attention the details.
ReflectionI'm looking forward to getting to a new chapter and to see some "other topics in group theory"
The most difficult part was the proof of theorem 7.52. It was long and I found it difficult to pay attention the details.
ReflectionI'm looking forward to getting to a new chapter and to see some "other topics in group theory"
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