In this paper we introduce the smallest equivalence relation ξ∗ on a finite fuzzy hypergroup S such that the quotient group S/ξ∗, the set of all equivalence classes, is a solvable group. The characterization of solvable groups via strongly regular relation is investigated and several results on the topic are presented.

The lifting of results from factor groups to the full group is a standard technique for solvable groups. This paper shows how to utilize this approach in the case of non-solvable normal subgroups to compute the conjugacy classes of a finite group.

We show that the epimorphism problem is solvable for targets are virtually cyclic or a product of an Abelian group and finite group.

We say that a group has property R∞ if any group automorphism has an infinite number of twisted conjugacy classes. Fel’shtyn and Gonçalves prove that the solvable BaumslagSolitar groups BS(1, m) have property R∞. We define a solvable generalization Γ(S) of these groups which we show to have property R∞. We then show that property R∞ is geometric for these groups, that is, any group quasi-isomet...

We give an example of a residually-p finitely generated group, that satisfies a non-trivial group law, but is not virtually solvable. Denote by Fn the free group on n generators. Recall that, given a group word m(x1, . . . , xn) ∈ Fn, a group G satisfies the law m = 1 if for every u1, . . . , un ∈ G, m(u1, . . . , un) = 1. Given a set S of group laws, the n-generator free group in the variety g...

Suppose that G is a finite group and x ∈ G has prime order p ≥ 5. Then x is contained in the solvable radical of G, O∞(G), if (and only if) 〈x, xg〉 is solvable for all g ∈ G. If G is an almost simple group and x ∈ G has prime order p ≥ 5, then this implies that there exists g ∈ G such that 〈x, xg〉 is not solvable. In fact, this is also true when p = 3 with very few exceptions, which are describ...

In 1964, John Stallings established an important relationship between the low-dimensional homology of a group and its lower central series. We establish a similar relationship between the low-dimensional homology of a group and its derived series. We also define a solvable completion of a group that is analogous to the Malcev completion, with the role of the lower central series replaced by the...