238 Dummit & Foote Abstract Algebra Ch 4 Reading 2 of 2

[00:00:00]So this is the second part of uh chapter 4 reading and the reason why I broke it off was not really because of section 4.4 it was because of section 4 4.5 on pseudo theorems and so 4.4 uh is a pretty straightforward read automorphisms. I'll show it in the book but I've already I've already had to think through what my plan is going to be for 4.5.

[00:00:25]Um yeah this this section is going to be special special in the amount of work that I devote to it I think because it will it will require it. Yeah. So automorphisms the automorphism group this gets used later uh a lot in uh in um gala theory.

[00:00:50]Yeah. all about conjugation and forming all the possible automorphisms and turning them into a group.

[00:01:02]Yeah, conjugate elements and conjugate subgroups have the same order. Of course, there's a problem like that in chapter 3 already as a sort of a harbinger of things to come.

[00:01:14]Yeah, the automorphism group of the psychic group of order N is isomeorphic to the multiplicative group Z and Z and a beginning group of order F ofN the totium to function oilers function.

[00:01:28]Yeah. But then yeah, this this this is a special I uh the entire proof [sighs] of seal theorem.

[00:01:41]I'm going to have to go through it again and again and again. Um the way everything got broken up into orbits.

[00:01:52]Yeah, this I read this multiple times.

[00:01:57]Uh, and I'm gonna have to I'm gonna have to do something special to truly understand what this is actually doing because as of now I do not. That is a problem for me. Now the other problem for me is uh the way the logic works with these examples of the applications of seal uh uh theorem which is really three theorems and Gallian treats them as one and other books break them up into separate uh theorems is Gallion uses the same logic but the way it's done in Gallian I really understood it a lot better. It just boils down to that. Yes. uh you're you're trying to figure out uh something about these groups and uh you're going to work through most of the time you're really working through uh let me show the part of this part. Okay. So most of the time you're you're mcking around with a simple calculation that will show you what n subp is. uh and then that tells you how many groups that there are there are and then also you play with that to figure out the orders to say okay uh if I use that set of groups uh that the theorem is saying could be there uh I run out of elements and so because I run out of elements then it cannot be that and so you're sort of con constraining what the possible configuration of these groups is by using Seal's theorem part three really this one. I mean, I know that's what the crux of the matter is, but when I read this, it just reads like mud. Mud. All of these read like mud.

[00:03:41]Uh, and so I'll have to read it again.

[00:03:43]And of course, I I'm very much looking forward to the challenge. It's like climbing Mount Everest of working through these 56 problems. And I do plan to attempt every single one of them. And I suspect that it will take me several weeks uh to go through these problems and I will do so. Yeah. So that's that's the just laying out why this is a separate video uh for my own uh just for my own path. All right. So automorphisms and that I already discussed in the last video.

[00:04:20]Then just continued on. This is all laying out things that will get used uh in galwa theory much later in this book and the same would have been in gallion.

[00:04:33]Just following through looking through the examples actually working them out.

[00:04:45]Yeah, just reading through everything was going just fine until I started reading this chapter until I started reading the proof of seals theorem. Yeah, that's where really uh started drawing the groups.

[00:05:07]Yeah, I'm gonna have to go through and I'm gonna have to find somewhere where orbits are discussed in a planer way and they're explained in a planer way. I know what they are uh because they're discussed in Gallian. And so really in the case of uh say a permutation if you have like 1 2 3 separate to four five six those are two orbits 1 2 and three are in an orbit and four five and six are in an orbit. So I get that and I get how the orbits combined with the um the stabilizers get combined into this orbit stabilizer theorem which is discussed in gallion and I don't know why it has not been discussed in dominant foot yet. I'm not going to look it up right now because I don't want to take time in this video where it is in dominant foot.

[00:06:03]Yeah, just the applications part of this chapter is just muddy in my in my in my opinion how from my reading. So all this all this I will have to read again. Now of course I did get my you know you get back on a bike uh and you can write it and I remember doing all these calculations in Gallian. So yeah once I got started with the calculations I could anticipate what uh dominant foot were going to say. I knew what these values work as I had calculated them like N5 uh in this one and then this one as well. Uh but but the way this logic is laid out, the early logic especially, okay, the early logic especially in this one, um I'm going to have to go back and really and I may have to actually reread all of this section uh and then draw sort of like diagrams for how the logic works out uh because of how the constraining is happening in uh in the examples when you go order by order with examples.

[00:07:13]Yeah.

[00:07:16]Yeah. This was very dense. Very dense. I just I laid it out. Okay. I read it.

[00:07:21]That's fine. But it's not fine. U Yeah.

[00:07:25]This will have to be all reread for sure. This was a very dense uh part of the of the book in my opinion.