Group Theory (Math) Question?

Question

What if the dot product is defined by the cross product, then "a dot e" is not equal to "e dot a", and then so are "Vectors" then not a group? Well then, if Vectors are not the elements of a/the group, then what are the elements of a group when taking the Cross Product?

Answer 1

The question "what are the elements of a group when taking the cross product" seems to presume that, given any operation, it is possible to make it the binary operation of some group. This is not the case. There are several problems with making the cross product the binary operation of a group. One is that the cross product is not associative. For example,



((1,0,0) x (0,1,0)) x (1,1,1) = (0,0,1) x (1,1,1) = (-1,1,0)

but

(1,0,0) x ((0,1,0) x (1,1,1)) = (1,0,0) x (1,0,-1) = (0,1,0).



[I chose the vectors (1,0,0), (0,1,0), and (1,1,1) almost at random--- it's certainly possible to find vectors u, v, and w that do satisfy (u x v) x w = u x (v x w), but most choices of u, v, w do not make this true. And we only need one example where it isn't true to know that x is not associative.]



Another issue is that there is not even a left identity element for the cross product--- that is, there is no vector u satisfying u x v = v for all v. (Clearly the zero vector cannot have this property, and if u is nonzero, then u x u = 0 is not the same as u.) And if there is no left identity element for a binary operation, there is no two-sided identity element for a binary operation, and no conceivable meaning that one can give to "inverses" with respect to this operation (as inverses are defined in terms of an identity element).



The lesson is that there are commonly used (and useful!) binary operations that do not easily fit into the framework of group theory. This is OK, though. Group theory is a very applicable subject, but it makes no claim to be a "theory of everything", and there is a lot of math that simply is not group theory. I hope this helped.

Answer 2

I think you are confusing terms. A cross product defined for a vector space over a field makes it an "algebra" not a group. A vector space is always a group to begin with. The group operation is addition of vectors as the operation is a group. Since addition is defined to be commutative, it's an Abelian group. It has other properties besides that.



In a group, you don't have to have ab=ba. If ab = ba for all elements in the group, it's called commutative or Abelian.



It might help for you to review the definition of a vector space at

http://en.wikipedia.org/wiki/Vector_space#Definition

and reask your question. I don't understand what it is that you are asking. Cross product is an operation beyond group theory, and I don't think it's helpful to try to force it into a group theory model. Someone who understands vector spaces and algebras better than I do may have some better insights into what value there would be in applying group theoretic techniques to vector with cross product as the operation, but I'm guessing that's not a useful direction. The cross product is used in many applications in physics and engineering, and those applications go beyond group theory in their use of cross products and even more advance operations such as curl and tensor products.

Answer 3

I responded to you question.