(no subject)

[pozorvlak]

I've been tutoring a high-school student for his Cambridge entrance exams recently. Yesterday, we had the following conversation¹:

Him: I've been reading Fraleigh's book this week.
Me: Sorry, never heard of it. Is that one of the set books or something?
Him: It's called "Introduction to Abstract Algebra" or something like that.
Me: Doesn't narrow it down much, I'm afraid. Have you got it with you?
[He hands it over]
Me: Hmmm. Sets, [flickflickflick] relations, partial orders, equivalence relations, looks fine... [flickflickflick] groups, permutations, cosets, all very standard stuff... [flickflickflick] vector spaces, rings, OK, so far so good... [flickflickflick] Sylow's theorems? That's a bit advanced for an introductory book, surely? [flickflickflick] Simplicial homology? WTF? [flickflickflick] Galois theory?
Him: Do you think that would be useful for my first year, then?
Me: I think that will be useful up to and including your fourth year.
Him: So if I can work my way through it this year...
Me: If you can work your way through it this year, I'll start taking classes from you. So, how are you getting on with it?
[Warning: Actual Maths starts here]
Him: Well, I'm up to the bit about groups, but I need to go over it again.
Me: Yeah, that's par for the course when you're learning from books, I'm afraid. Anyway, groups are interesting because they capture the notion of symmetry in a very general way. You can take any object of any type, and take the transformations that leave it essentially the same, and they will form a group.
Him: What do you mean, "transformations"?
Me: Well, that depends on the type of object. Have you got as far as group homomorphisms?
Him: I've got as far as isomorphisms...
Me: [thinks: wtf?] OK, well a homomorphism is [scribbles definition], and an isomorphism is a homomorphism f such that there's another homomorphism g which is its inverse. [scribbles down, accompanied by "Moral Definition"]
Him: Why have you written "Moral Definition"?
Me: That's because there's an immoral definition, which you sometimes see, which works for groups and rings and things but not more generally. [scribbles down "Immoral Definition: an isomorphism is a bijective homomorphism".] Hmmm, I wonder what Fraleigh uses... [flickflickflick] Aha. "An isomorphism is a bijective homomorphism." Sod. [horrible suspicion starts to dawn...] Let's look at the index... [flickflickflick] "cartesian product, cases: argument by, centre" - oh God, he's written about homology without using categories. In [flickflickflick] 2002! Why would anyone do that? Arararargh!

Anyway, we spent most of the rest of the class talking about field extensions and Galois theory (as examples of places where equivalence relations and groups were useful): I don't know if he understood any of it, but I had fun :-)

¹ Lightly edited: we may not have used precisely these words, and the conversation didn't exactly happen in this order, but everything above was said at some point :-)

Comments

[michiexile]

Yeah, Fraleigh is kinda light on the modern-and-useful stuff. I remember reading parts of it, but this was before I grew up and learned Mathematics the way it should be done, so I didn't think much of it.

There is a bunch of authors out there, alas, who view categories as evil, spooky and icky, and therefore will go to insane extremes to avoid mentioning them, even if they end up doing things where the language is trivial and obvious with categories and awkward and bad without.

[pozorvlak.livejournal.com]

I hadn't come across it before - I used my Dad's copy of Birkhoff and MacLane when I was an undergraduate :-)

There is a bunch of authors out there, alas, who view categories as evil, spooky and icky
I know - depressing, isn't it? I believe that's true of a large fraction of mathematicians in the USA (outside Chicago, of course). I appreciate that the stuff actual category theorists do is highly esoteric and hard to get into, but that's true of most disciplines - the obscurity of modern group theory doesn't stop people using groups elsewhere in mathematics! But then, groups have had 150 years to catch on... I wonder if in 1907 mathematicians would go to insane lengths to avoid using groups in their discussions of symmetry?

[michiexile]

You, friend, need to sit down and read the Unapologetic Mathematician for therapy. Do that, and breathe deeply. You'll soon forget, temporarily, all the ugly bad icky scary mathematicians out there.

[(Anonymous)]

What books on modern algebra would you recommend, then?

[pozorvlak.livejournal.com]

Dunno, to be honest - I didn't use books all that much. Birkhoff and MacLane is still excellent, though. Lawvere's Conceptual Mathematics?

[pozorvlak.livejournal.com]

I don't mean to say that Fraleigh's book is bad - as I said, the wide range of material it covers means you'll get quite a lot of use out of it! A more categorical presentation of the material would (in my humble but correct opinion) make it clearer and simpler, but that doesn't stop it from being a basically good book, and the bits I saw looked fine. Though I'd avoid it for learning homology theory, I think!