1
$\begingroup$

This question may be silly to experts, but I am waiting for a response sir.

My question is

" Is there any existence of generalized Cantor's counting principle ( i.e the theory that decide whether a set is countably finite or not) to apply for groups ? "

So we know that there is an underlying set in every group, so can we apply the Cantor's argument for groups? .

Background: My plan was to count the cardinality of Tate-Shafarevich group by using the Cantor's theory, but generalizing Cantor's theory to apply for all sorts of algebraic structures may not be possible, moreover the Tate-Shafarevich group contains the Homogeneous spaces that are not simple sets, and also there is no proper group structure for Tate-shafarevich group except in the case of Pell-conics, so generalizing the Cantor's argument is very difficult for homogeneous spaces, but I think one can achieve it, by generalizing I think so. But is there any work in that direction? . Sorry if my question was bad, it was just a intuitive doubt, but it may seem silly to experts. Sorry

And in addition to it, please tell me the standard criteria or procedures that are used to determine whether a group is finite or not.

Thank you.

1 Answers 1

3

Cantor's theory does not provide a way to "decide whether a set is finite" (or countable, or uncountable) or not. Whether decision is interpreted to mean a decision algorithm or a proof in a particular axiom system, it has been proven by Turing and Goedel respectively that this is impossible for concretely specified sets, such as the set of outputs of specific computer programs that can be explicitly written down. It is impossible, in the same sense, to decide whether Diophantine equations in several variables (I think 9 are enough) have integer solutions, or a finite number of solutions, or to list all the solutions when the number is finite, or to determine any nontrivial property of the equation.

For groups, semigroups, and many other algebraic structures, it is not possible either to decide algorithmically, or to prove logically for any given concretely presented group, that it has a finite number of elements, or to determine whether the group has more than one element, or any other nontrivial structural property of the group. This theorem for finitely presented groups (a finite list of generators and relations) was proved by Boone, Novikov, Adian and Rabin.

For this and other reasons, there is no "standard criterion to decide whether a group is finite or not". You can show it is a subgroup or a quotient of a finite group, or isomorphic to a known finite group, or something close enough to a known group that it is possible to see the finiteness, or an extension of one finite group by another. Beyond obvious statements of this type, where finiteness is given for a basic list of groups and some other finite groups are created from those, or the analogous idea with infinite groups, there is no general method for taking a given group and mechanically determining whether it is finite or not.

  • 0
    I'll comment that, for diophantine equations, degree 4 diophantine equations are equivalent to turing machines. You may need an arbitrary number of variables, but you can reduce it to degree 4 because any natural number can be written as the sum of four squares (degree 2), and a multiplication relation can be written a = bc, so if b and c can both be written in degree 2 then the result is a 4th degree expression. Anyway, I don't remember any specific variable count threshold.2011-11-14
  • 0
    @Carl, that and not the number of variables is what I had in mind when saying 4 is sufficient. Also, the correct invariants are probably not number of variables but dimension and degree of a Diophantine system when considered as an algebraic variety over the complex numbers. I think the bound for either the dimension or the number of variables was 9 at one point but do not remember which.2011-11-14
  • 0
    -1. Re "it is not possible to decide algorithmically, or to prove logically for any given concretely presented group, that it has a finite number of elements": You have your quantifiers mixed up. What you meant is that there is no algorithm that always stops after a finite time and that works for any group, deciding whether it's finite. But for a given concretely presented group, it certainly _is_ possible to prove that it has a finite number of elements. Besides, all this is off-topic, because we certainly do expect to prove one day that sha is finite. Only Cantor has nothing to do with it.2011-11-14
  • 0
    @Alex B, although you correctly described the intended meaning, I think the written expression of that meaning was not "mixed up" (or much different from the expression in textbooks, papers, and lectures). I gave examples in the third paragraph of ways to "decide algorithmically, or prove logically" for *some* given groups that they are finite, and defined the impossible thing as "general method for taking a given group and mechanically determining whether it is finite or not". The latter is identical to your description, an algorithm that works in finite time for any finitely presented group.2011-11-14
  • 0
    @Alex B, by the way, what is meant by the quantifiers in your statement "for a given concretely presented group, it certainly *is* possible to prove it has a finite number of elements"? There are specific finitely presented groups for which proving they have a finite number of elements is equivalent to demonstrating the consistency of a particular axiom system such as ZFC. This is a special case of the Adian-Rabin theorem.2011-11-14
  • 0
    @zyx I guess we are having trouble with deficiencies of the English language. I meant that for some given presentations, it is possible to prove finiteness. Of course not for all, since some of them will in fact not be finite. Your post can be read as saying that "it is not possible for any concrete presentation". Anyway, I think this is clarified.2011-11-14
  • 4
    But I still think that the answer is off topic. The OP asked whether one could prove finiteness of sha using Cantor's diagonal argument. This reveals complete confusion about Cantor's argument, about the difficulties in dealing with sha, and possibly some other rather fundamental things, and your post didn't address any of them.2011-11-14
  • 0
    @AlexB. :sir if the answer was insufficient why can't you write up a new answer if you are free sir, I have even followed all the things you have said, so please do answer stating any known procedures to find whether a group is finite or not, thanks a lot sir2011-11-14
  • 0
    @zyx:Thank you sir, let me wait for an answer which fixes and covers all points, anyway '+1' for your efforts sir2011-11-14
  • 0
    @Alex: the OP did not mention Cantor's diagonal argument, and Sha was discussed only as "background" (in which it is unclear to me what the goal is). The question in the posting is highlighted, in its own box, introduced with the words "my question is", and makes no reference to Sha. I read it at face value, and the answer is that the OP is asking about importing a certain type of capability from sets to groups, but that capability did not actually exist for sets, and does not exist for groups either.2011-11-14
  • 0
    Oh,thanks a lot sir now my doubt got fixed @zyx2011-11-15