I don?t know anything about group theory but I have scanned an online text (incomprehensible) and it seemed that there was some similarity to vector spaces. Is this true? And could someone recommend a text that is written with the simpleminded in mind? I just want to get a feel for this stuff as it is used in physics. It would be great if there was no math involved. ‘:D’
A vector space is an extension of a group; in other words, to get a vector space, you start with a group and add more stuff to it, basically.
I’m not sure of a good text to recommend; what are you looking for? More group theory oriented, or more vector space? Or just algebra in general? The book used in the first algebra class I ever took was Modern Algebra: An Introduction by Durbin:
No vector space stuff that I recall, but groups, rings, fields, and that sort of thing. If you don’t have much of a math background, it may take a little work to read, but I think it’s pretty clear, as math textbooks go.
Herstein’s Topics in Algebra is one of the standard introductory texts, but is probably a little harder to read than Durbin (but still do-able, if you stick with it).
Well, they are both axiomatic algebraic systems, so they are certainly similar to that extent. There is also a more concrete connection: in any vector space the vectors form an abelian group under addition.
Group theory is used in particle physics, I understand, so perhaps you might be able to find suitable material by exploring that avenue.
And I could swear I had included a link to Herstein, too. Well, here it is:
From what I know of vector spaces (Anton and Marcus&Minc) it appears a bunch of stuff was developed for an Euclidean inner product space and then the rules for the inner product were axiomatized and applied to all kinds of other stuff such as matrices, polynomials and differential equations etc. And even though I have no idea what the norm for instance would mean for these other things it still seemed pretty neat.
This page says, “a group G is a set with a rule for assigning to every ordered pair of elements a third element satisfying (4 rules that I can’t make the symbols for).” These seem like some of the same rules for an inner product space.
So are you guys saying that you can get to a vector space from the other direction i.e. starting from a more abstract principle and moving in reverse?
And what do you think of the text in the link?
A group operation is completely different from an inner product. For one thing, in a group, the two group elements get mapped to another group element; with an inner product, two vectors get mapped to a scalar, not another vector. I mentioned earlier that a vector space is a group with other stuff tacked on (I forgot to mention, and Jabba reminded me, that it has to be an abelian group).
As Jabba also mentioned, it’s the vector addition in a vector space that you should be thinking of as corresponding to a group (since it is an (abelian) group), since, going by your link:
1.A.1 The addition of two vectors is still a vector.
1.A.2 Vector addition is associative.
1.A.3 The zero vector added to any other vector just gives us that vector back.
1.A.4 Any vector v plus (-1)v gives us the zero vector.
In a vector space, we also have:
v+w = w+v
for any vectors v and w, and this is what makes it an abelian group (the operation commutes).
I don’t think that’s what I’m saying. If I understand you right, I think you’re asking if starting with a more abstract principle (a group), can you get to a vector space.
Well, like I said, a vector space is an abelian group with (much more) additional structure, but I wouldn’t say that a group is any more or less abstract than a vector space, they’re just two different algebraic structures. If you feel more or less comfortable with vector spaces, you should be able to learn some group theory without too much trouble.
About your link, if you’re really looking to learn group theory, I don’t think that book would be the right approach for that–it’s not really a group theory book, leaves a bunch out, and breezes through the rest. As your first exposure to group theory, I’m not sure that you’d really be able to get much group theory out of it.
However, if you just want a brief taste of group theory, then get right into the physics applications, it may be a good choice; I don’t think I’m qualified to judge it, though, since I don’t know a hell of a lot of physics; I’ll leave that open to one of our physicists here.
I wouldn’t recommend that. While it’s true that some groups are used extensively in particle physics, the particle physics books generally assume that you already have the appropriate background in group theory, and there’s a lot of group theory which doesn’t make it into particle, at all.
Sorry I meant to say an inner product space or a vector space with an inner product.
Actually now that I review my prior post I did say
Or am I confused? Is an inner product space not a vector space?
An inner product space is a vector space with (get ready) an inner product. Not all vector spaces have inner products defined on them, although there’s no reason why you can’t define one.
For an introduction to group theory, you could do worse than Herstein, but you could also do better. Go to amazon and look at some of the texts recommended by reviewers of Herstein’s book.
Aargh! What confusion.
Think of a group as a set of permutations that includes the identity permutation (that doesn’t do anything) and “closed” with respect to inverses and composition. (so if p and q are two permutations in the set, so is p followed by q, usually denoted qp) as well as the inverse of p (and of q). Now the set of all vectors of some dimension (which may be infinite) are actually a group. They permute the whole set of vectors. An individual vector v permutes the set by adding v to each element. Adding v and then w is the same as adding v+w and adding -v undoes the effect of adding v. Also adding the 0 vector does nothing. A vector space may additionally have an inner product and the ones that are most useful in physics do.
I first learned about all this from Birkhoff & Mac Lane, long out of print and the successor, Mac Lane & Birkhoff is doubtless too abstract. As are most introductory algebra texts.
It is, in any case, rather misleading too think of a vector space as a group. It is, but it is so much more than that that the theory of vector spaces (linear algebra) is essentially disjoint from the theory of groups. Actually, the Schaum Outline on Linear Algebra is not bad for that.