A Generalization of Cantor's counting theory 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.
 A: 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.
