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Abstract Algebra 3: the Revenge of the Groups

USA | Wednesday, 10 June 2015 | Views [169]

Today felt less busy than yesterday. We started later, but ended at nerly thie same time, which was part of it. However, I think the bigger difference was that today made more sense than yesterday. It was also more focused, with everything being about algebra. None of the mix of analysis and algebra and topology we'd gotten yesterday.

The fist part of the day was the Algebra Short Course. Like Analysis, it aims to give us a brief overview of material that would be covered in more detail in undergrad and graduate courses. Like Analysis, we were also given lecture notes. These were typed out. As I was skimming through to see what we'd cover today, I was subconsciously comparing it other LaTeX documents. Generally, it feels like every professor (and, for the most part, students as well) have slight differences in template or formatting choices. It's kind of fun to try and identify them. (I have an odd sense of amusement.)

We got through a more thorough introduction to groups before moving on to discuss subgroups. We proved some propertie, and took others on faith and intution. As with analysis, most of it was review of material I'd already learned in much more detail. There was an interesting diversion into groupoids (sets and binary operations which may or may not have some of the properties of a group.) I'd listened to a professor give a talk on loops last semester, and I'd found his talk interesting, and would like to learn more about them at some point.

But for today, we mostly just reviewd groups. Being back in a classroom learning abstract algebra, I defaulted to last semester and returned to my final project for that class. I turned it in weeks ago, but there's a part of my brain that's still trying to answer a question I posed in it. That project took all the time I gave it and, greedily enough, it still wants more. I feel like I'm getting close to being able to answer it, though I'm not sure what I'd do if I succeed.

Unit graph of Z_5 direct sum Z_5, if that means anything to you

Isn't it beautiful? Who wouldn't want to spend all their idle time staring at pictures, I mean, proving properties, of graphs like that?

We discussed subgroups during the short course, but didn't have time to get to normal subgroups. Which was unfortunate, because we needed that during the research seminar for further explanation of group presentations. So we got a brief explanation of those, but will probably go more in-depth of Thursday during the next algebra class.

Beyond the definition of normal subgroups I was already familiar with, he introduced a new term- normal closure of S in G. Essentially, it's the intersection of al normal subgroups containing a set S. It's also necessary terminology for a certain definition of a group presentation. Not the definition I'd seen before and was working off, but a certain definition. (Hopefully tomorrow we'll get to why the two are equivalent.)

We did get around to proving that every group is isomorphic to a group determined by its presentation. Which is an important theorem that I believe would be true. After talking it through with Angelo after class, and getting a few more terms defined, I even mostly understand how we proved it. If I believe a previous theorem that he stated without proof or explanation, then I see how the second theorem follows from it. But without having any idea where the first theorem came from or why it would be true, I'm not quite able to do that.

By this point, he was mostly out of time, but the professor wanted to return to an earlier example and show us how this might work. Except it didn't, really. Now we were completely out of time, but he wanted to give us a more satisfactory explanation. It still didn't work. Now he was keeping us a few minutes late, (it's not like we had anywhere else to go) but we still weren't there. He could now see the steps, but, since we were out of time, he gave us a lemma to prove and an exercise to think about tonight. And then he let us go.

We went back to the dorm. I relaxed for a bit on my own, then noticed that Dana, Griselda, and Angelo were in the courtyard between the buildings (and in clear sight and hearing range of my window) talking over textbooks. So I went out to join them and discuss questions from last night and classes earlier today. Then it was nearly 6:00 and time for our review session with Hannah, so I ran to the Armstrong Center to grab food before it closed and I have money go to waste. (I still ended up wasting 32 cents. Which was much worse than yesterday, where I only wasted 5. I could have cut it down to 1 if I'd trusted my arithmatic over the cashier who said tax counted extra.) Then I went back to the dorm for the review session.

For me, it was a review of a review. Hannah spent more time explaining nomral subgroups, and I spent more time thinking about rings and zero-divisors. And then at the end, she explained what a diagram commuting meant and, in the course of giving an example ended up with an informal proof of the lemma from classs. So that was nice.

Unfortunately, it's been another day, and the research seminar professor still hasn't given us the problem (problems?) I understand that he wants to make sure we have the background knowleedge before diving into the problem, but I feel like I'd appreciate the background more if I understood how everything was supposed to fit together. Hopefully we'll find that out tomorrow.

Tags: abstract algebra, graphs, groups, math

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