Applications of hypergroups have mainly appeared in special subclasses. One of the important subclasses is the class of polygroups. In this paper, we study the notions of nilpotent and solvable polygroups by using the notion of heart of polygroups. In particular, we give a necessary and sufficient condition between nilpotent (solvable) polygroups and fundamental groups.

The aim of this paper is to prove Theorem 1.1 below, a generalization to virtually nilpotent spaces of a result of Wilkerson [13] and Sullivan [12]. We recall that a CW complex Y is virtually nilpotent if (i) Y is connected, (ii) 7r~ Y is virtually nilpotent (i.e. has a nilpotent subgroup of finite index) and (iii) for every integer n > 1, zr~Y has a subgroup of finite index which acts nilpoten...

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield–Kan homology completion tower zkX whose terms we prove are all X–cellular for any X . As straightforward consequences, we show that if X is K–acyclic and ...

Let C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C are torsion-by-nilpotent, then all soluble groups of C are torsion-by-nilpotent. From that, we deduce the following consequence, similar to a well-known result of P. Hall: if H is a normal subgroup of a group G such that H and G/H ′ are (locally finite)-by-nilpotent, then G is (l...

A nilpotent group G is a finite group that is the direct product of its Sylow p-subgroups. Theorem 1.1 (Fitting's Theorem) Let G be a finite group, and let H and K be two nilpotent normal subgroups of G. Then HK is nilpotent. Hence in any finite group there is a unique maximal normal nilpotent subgroup, and every nilpotent normal subgroup lies inside this; it is called the Fitting subgroup, and...