Normal Groups, p-Sylow Groups, Solvable Group ... Oh My!

So, I've been studying a lot of Algebra lately. Currently we're working on Sylow groups and just lat class we started proving 'solvability' and 'simplicity' of groups using the Sylow theorems. And, shocker, since I've never seen this stuff before, I have double the work of not only trying to understand the concepts of these theorems and properties, but also learning the basics of the concepts. Which is fine, I'm doing okay with it - surprisingly. Right now, I'm working on proving that any group with order less than 60 is solvable. I've gotten it down to about 15 groups, whose orders don't follow the form of a p-group (i.e. have order p^n for some p prime) or the form pq, where p and q are distinct primes. I have a hint from a fellow first year to consider looking at the theorem that we have which stats that G is solvable if and only if H and G/H are solvable (where H is a normal subgroup of G). So, that's my current status.

We just started measurable functions in my Real Variables class. That stuff seems to be going okay, at least for right now. I didn't freak out of the homework that was due yesterday which is a massive bonus. I know I botched one of the proofs; the one proving that even under a continuous mapping, you can map a measurable set into a nonmeasurable set. We did this using the Cantor Lebesgue function, the function that is commonly used in tandom to describe the devil's staircase (i.e. maps intervals of [0,1] which have been removed forming the Cantor set to constants, with linear mappings on intervals that remain). So, I recognized most of what I needed to, but there was this one part actually showing the nonmeasurability of a subset of [0,1]. I just used the theorem we had to state that such a subset exists. So, I'm not completely and utterly distraught over that error. And the second error that I'm pretty sure I made, was in proving that if x is an element of the Cantor set, x can be written in base 3 using only 0's and 1's in its ternary expansion. That one was just a really sloppy proof. I was not proud of that one at all. But, on lighter notes! I think I pretty much nailed the other 5 problems in the set, so yay for that!

And Algebraic Topology, I needn't talk about. As bad as it is on my part, I haven't touched it in quite some time. That'll need to change, because our first assignment is due a week from today, and none of us know anything that we've been doing in that class. So, we'll be having an Algebraic Topology Library party, I'm sure. These are always fun. We just grab any book we think might be useful to us, get a big table, and go at it. It's one of those typical things you see in the movies (which I never thought actually happened) with the mass chaos of students with every book opened all across the table. It's quite funny actually.

Well, on that note, I must get to work. I think I'll top thinking about that proof that any group of order less than 60 is solvable for the moment and move on. It's only the first of 10 problems. And maybe the proof of it will come a bit later - I love when that happens. You don't understand part of a proof, and then BAM! The proof is completely laid out in your head. It's a wonderful experience, and I hope any mathematician reading this has had that joy. And I'm sure it applies to other disciplines as well in some way or another.



*UPDATE: So, I just went back to do the next problem in my problem set for Algebra, and came across two more things (that I will need to prove now) that help me in getting through to the goal of proving that any group of order less than 60 is solvable! Namely, now I need to prove that any group of order (p^2)q, where p and q are distinct primes is solvable, and that any group of order 2pq is solvable, where p and q are distinct odd primes. So, now I'm down to groups of order 36, 40, 48, and 56. Back to work!