The notion of free group is defined, a relatively wide collection of groups which enable infinite set summation (called ), 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 kind of inverse limit...

Homology functor in the spirit of the AST is defined, its basic properties are studied. Eilenberg-Steenrod axioms for this functor are formulated and established.

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

We shall show that there exist sofic groups which are not locally embeddable into finite Moufang loops. These groups serve as counterexamples to a problem and two conjectures formulated in the paper by M. Vodička, P. Zlatoš (2019).

A group $G$ has the endomorphism kernel property (EKP) if every congruence relation $\theta $ on $G$ is the kernel of an endomorphism on $G$. In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.

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