That sounds like Russel's paradox of the infinite sets. That you can just keep grouping an empty set inside another ad naseum. But in your case think about taking an element (say {0}) of a subgroup of a subgroup of a subgroup... etc. out of that subgroup, and diminishing the amount of sets containing that element until it only contains itself.
Yes, you would eventually get to the purpose of the group, or initial element, which is {0}. If I linked it correctly, asking "why" is just breaking down the containers human use in language to the point where you answer the question in terms of unabridged intuition. Well, intuition as in a Kantian model, anyway. It could just as well be highly technical language, but it still erodes away unnecessary thought processes.
Oh. Goddamnit. I didn't even add anything worthwhile to that.