Recent content by CoCoA

What is a good way to describe and count the Sylow 2-subgroups of S5? The ones isomorphic to D8 should be simple enough to describe as the extra conjugates of the Sylow 2-subgroups of S4, but I am trying to figure out how to easily describe any/all those isomorphic to Q8, which don't exist in S4.

Yes, that is a basic characteristic of Groebner bases; you can find in the book by Backer & Weispfenning referred in Faugere's paper or many other introductions to Groebner bases. If you need brushing up, you should look at one of these before getting too deep with a particular algorithm like F4.

Although that is quite valid, in this problem I would find it simpler to show that G spans P3 and thus is linearly independent.

I am frustrated with what I had to do to decipher it. It should have been written more clearly. One would expect the rows of A to be in the order f_3, bf_4, df_4 sicne that is the order of the set given above the matrix. However, f_3 has two terms each in common with bf_4 and df_4, just as...

I'm afraid I do not know a lot about this algorithm, but I believe that G={f_1,f_2,f_3,f_4} before the second while loop is entered. The first while loop does not stop until F is empty. It is also pretty clear that the labelling of f_1 to f_4 are backwards in the paper on page 14 -- that is, f_4...

Most of the time in an introductory course, the non-trivial subspaces will be infinite, because they are usually over the filed of Real numbers or Complex numbers, but there are also lots of finite fields and any finite-dimensional vector space over one of those will be finite, so all of its...

I hope the original poster doesn't mind my suggesting reading one of those books that teaches one how to prove things, basics of set theory, other important ideas used in abstract and proof-oriented math courses. They can provide, in a sense, the "rules of operation". It can be read parallel to...

I'm not sure without your definition of your set U. On the face of what you wrote, I am concerned that you give (-x2-b,-x2) in U; here we have x1=-x2-b which is not equal to -x2+b as in your condition on elements of U. In fact, if v and -v are in U, then 0 is in U by additive closure. Thus...

Pick a vector v in U. By the additive closure, we know rv in U for each integer r. What about xv for some non-integer scalar x?

I would first think about what condition of being a subspace is going to be violated by the subset. It is closed under addition by your problem statement, so your subset must not be closed under scalar multiplication.

You can absolutely do it if you want to. You may take more time than the average student, or you may not be able to work full time the whole way through the program; this can be a difficult matter for someone with family responsibilities. But hard work and persistence matter more than just about...

Don't ask me that one, ask your parents.

I would never be a high school teacher, but if I were and a student called me by my first name instead of using Mr., I would probably throw the student out of class. (Not because he/she is a student, but because he/she is a non-adult addressing an adult - and an authority figure at that - in a...

I think schools have a wide range of effectiveness in their technology initiatives. For example, some schools have implemented well-developed online courses, others (such as one I work for, that I will not name) has several online classes that I candidly believe are nothing more than glorified...

I took an line course a few years ago with recorded video lectures online. It was actually pretty good. On the other hand, about a year ago I was "talked into" (forced into?) teaching a course onlilne that I thought wasn't that amenable to online delivery. It turned out to be pretty bad.