The authors prove that a local n-quasigroup defined by the equation , where , i,j = 1,...,n, are arbitrary functions, is irreducible if and only if any two functions and , i ≠ j, are not both linear homogeneous, or these functions are linear homogeneous but . This gives a solution of Belousov’s problem to construct examples of irreducible n-quasigroups for any n ≥ 3.
We are investigating quasigroup functional equation classification up to parastrophic equivalence [Sokhatsky F.M.: On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian)]. If functional equations are parastrophically equivalent, then their functional variables can be renamed in such a way that the obtained equations are equivalent, i.e., their solution sets are equal. There exist five classes of generalized distributive-like quasigroup...
A Fitting set is called elementary if it consists of the subnormal subgroups of the conjugates of a given subgroup. In this paper we analyse the structure of the finite solvable groups in which every Fitting set is the insiemistic union of elementary Fitting sets whose intersection is the subgroup 1.
We characterize the solvable groups without infinite properly ascending chains of non-BFC subgroups and prove that a non-BFC group with a descending chain whose factors are finite or abelian is a Cernikov group or has an infinite properly descending chain of non-BFC subgroups.
Let be a group and let be a finite subset. The isoperimetric method investigates the objective function , defined on the subsets with and , where is the product of by .In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtain generalizations and short proofs for several known results. We also give some new applications.Some of the results obtained here will be used in coming papers to improve Kempermann structure...