The notion of free group is defined, a relatively wide collection of groups which enable infinite set summation (called commutative $\pi$-group), is introduced. Commutative $\pi$-groups are studied from the set-theoretical point of view and from the point of view of free groups. Commutativity of the operator which is a special kind of inverse limit and factorization, is proved. Tensor product is defined, commutativity of direct product (also a free group construction and tensor product) with the special...

The isomorphism between our homology functor and these of Vietoris and Čech is proved. Introductory result on dimension is proved.

The main result of this note is that a finitely generated hyper-(Abelian-by-finite) group $G$ is finite-by-nilpotent if and only if every infinite subset contains two distinct elements $x$, $y$ such that ${\gamma }_n\left(〈x\text{,}{x}^y〉\right)$$={\gamma }_{n...}$