Two Very Early Problems, a Simple Solution, and a New Problem



As an undergraduate student whenever I studied some subject I tried to come up with problems. Many of these problems were artificial or silly and, of course, I forgot most of them. But a few still make sense. Here are two problems: 

1) Let B be the unit ball (or the unit cube) in R^d. Does every function from B to B which is the differential of a real function on B have a fixed point?

2) Is there a common generalization for Sylows’s theorem and Frobenius’s theorem?

Updates:A few typos corrected; thanks Lior! A remark by David Speyer suggests that both results we would like to find a common generalization to, are due to Frobenius. (Who was motivated by Sylow’s theorems.) Emmanuel Kowalski’s was partially motivated by these problems to present an old sporadic problem of his. (It is a mystery for me why his post is not mentioned as a track-back in this post.) 

1) A fixed point theorem for differentials?

One of the delightful theorems you learn in the first year of undergraduate studies (and later teach as a TA, and later teach as a professor) is the intermediate value theorem. If a continuous function satisfies f(a)<0 and f(b) >0 then for some point c in the interval [a,b], f(c)=0. Later you learn Darboux’s theorem asserting that if f is a differential function, and f'(a)<0 and f'(b) >0 then for some point c in the interval [a,b], f'(c)=0. Your first reaction is: “Of course, f'(x) is continuous and we can apply the intermediate value theorem.” But, no, you soon learn some subtle examples where the derivative is not continuous!

One equivalent formulation of the intemediate value theorem asserts that every continuous map from [0,1] to itself has a fixed point. Namely, for some x \in [0,1], f(x)=x. This is a special case of Brouwer’s fixed point Theorem (which you learn in the second year) which asserts that every map from the unit ball of R^n to itself has a fixed point.

We can ask if every function from the unit ball to itself, which is a differential of a real function on the unit ball, has a fixed point.

There are many very interesting and difficult problems related to basic real analysis. Not many mathematicians are fully aware of the rich and beautiful modern results in real analysis. (Just like many people outside mathematics are not aware that there is more to be discovered in mathematics itself.)

For example, it has only recently been proved by Csörnyei, O’neil, and Preiss and by Elekes, Keleti, and Prokaj, that the composition of derivatives of differential functions has the fixed point property. This is not easy at all. Also, the question regarding connectivity of the graph of differentials of functions was studied extensively. See this paper by Csörnyei and  Holický

In Budapest, I mentioned this problem to Miklos Laczkovich. (His UCL home page mentions a few open problems in real analysis.) He asked Marton Elekes (an author of one of the papers I mentioned above, and the son of György Elekes whom we mentioned in connection to product sum theorems). Elekes found a simple proof that the answer is yes – there is always a fixed point . Suppose you want to prove it for a function f(x,y) whose derivative maps the unit square into itself. What you need to do is to inspect the behavior of f - x^2/2 - y^2/2 in the boundary of the square.

So this problem was not so good, but the following problem proposed by Laczkovich might be. 

Problem: Let X be a set homeomorphic to the unit ball in R^d. Does every function from X to X which is the differential of a real function on X have a fixed point?


2) Joining Frobenius’ and Sylow’ theorems

Sylow’s theorems in group theory, which we studied in the second year of undergraduate studies, always seemed to me as one of the few theorems I did not have a conceptual understanding of. This makes Sylow’s theorems rather mysterious and charming. (A similar impression with the opposite reaction is expressed by Tim Gowers in this interesting post.) 

Sylow’s theorem (one of them) asserts: In a group whose order is divisible by p^i there are 1(mod p) subgroups of order p^i.

Frobenius’ theorem asserts: In a group whose order is divisible n, the number of solutions to the equation x^n=1 is zero modulo n.

 (Frobenius was probably inspired by Sylow.)

Sylow intersection with Frobenius: The case i=1 of Sylow’s theorem is the same as the case n=p of Frobenius’ theorem.

Is there a nice (Sylow JOIN with Frobenius) theorem? The case i=2 of Sylow’s theorem is the place to start.

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9 Responses to Two Very Early Problems, a Simple Solution, and a New Problem

  1. Lior Silberman says:

    Shouldn’t orders of the groups be divisible by p^i and n, respectively?

  2. David Speyer says:

    An interesting hint on the Sylow problem: The part of Sylow that you are using is:

    If p^k divides |G| then the number of subgroups of order p^k is 1 \mod p.

    According to Jacobson, Basic Algebra I, p. 81, this result is actually due to Frobenius! Based on skimming Sylow’s paper, it looks like he only proves this result when p^{k} is the greatest power of p dividing |G|. It might be worth tracking down Frobenius’ original paper to see whether he makes a connection between his results.

  3. Gil says:

    Dear David, That’s interesting. Indeed I did not find this stated as part of the 3 Sylow’s theorems in Marshal Hall’s book and counted on my memory. So it is quite possible that my question is actually about finding an interesting common generalization of two results by Frobenius himself, and that both these results were inspired by Sylow’s theorem.

  4. Actually, I think my blog is set up so that it does not do automatic trackbacks; I’ll change this setting since I’ve started linking to other posts more frequently.

  5. Pingback: The Intermediate Value Theorem Applied to Football « Combinatorics and more

  6. chandrasekhar says:

    There is also this Banach contraction theorem, which talks about fixed points in Complete metric spaces, provided the map is a contraction.

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  8. Ben Wieland says:

    Here is Frobenius’s 1903 paper on his theorem. It starts by citing his 1895 generalization of Sylow’s theorem, but I was not able to find a digital copy of that.

    Here is a 1988 paper by Brown and Thévenaz that proves a composite generalization of the Sylow theorem. They then deduce the Frobenius theorem by combining their congruence with Möbius inversion on the poset of subgroups.

    • Gil Kalai says:

      Dear Ben, many thanks! This results by Brown and Thévenaz is certainly in the spirit of what I was thinking about and I look forward to try to understand the statement.

      I suppose the one possible generalization I was hoping to state would deal with systems of equations that are modeled after a system of generators/relations describing p-groups of order p_k, where you replace appearances of p^j by n^j. Another direction was to replace Frobenius theorem that gives a congruence relation for number of cyclic groups of orders dividible by n, with similar congruence relations but not just for cyclic group.

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