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.