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.

We study two group theoretic problems, Group Intersection and Double Coset Membership, in the setting of black-box groups, where Double Coset Membership generalizes a set of problems, including Group Membership, Group Factorization, and Coset Intersection. No polynomial-time classical algorithms are known for these problems. We show that for solvable groups, there exist efficient quantum algori...

We construct a solvable group G of order 5 648 590 729 620 such that the set of element orders of G coincides with that of the simple group S4(3). This completes the determination of finite simple groups isospectral to solvable groups.

We show that the right ideal of a Novikov algebra generated by square nilpotent subalgebra is nilpotent. also prove [Formula: see text]-graded text] over field with solvable text]-component solvable, where finite additive abelean group and characteristic does not divide order text]. any automorphisms if invariants solvable.

We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.

The paper is devoted to an application of Lie group theory to differential equations. The basic infinitesimal method for calculating symmetry group is presented, and used to determine general symmetry group of some differential equations. We include a number of important applications including integration of ordinary differential equations and finding some solutions of partial differential equa...

We prove that an element g of prime order > 3 belongs to the solvable radical R(G) of a finite (or, more generally, a linear) group if and only if for every x ∈ G the subgroup generated by g, xgx is solvable. This theorem implies that a finite (or a linear) group G is solvable if and only if in each conjugacy class of G every two elements generate a solvable subgroup.

A difficult problem in the character theory of solvable groups is to show that the number c.d. (G) of irreducible character degrees of a solvable group G is equal to or greater than d.l. (G), the derived length of G. Isaacs [4] has shown that d.l. (G) ^ 3 c.d. (G)-2 for every solvable group. Berger [1] subsequently proved that d.l. (G) < c.d. (G) when | G | is odd. This problem belongs to the c...

‎the concept of the bipartite divisor graph for integer subsets has been considered in [‎m‎. ‎a‎. ‎iranmanesh and c‎. ‎e‎. ‎praeger‎, ‎bipartite divisor graphs for integer subsets‎, ‎{em graphs combin.}‎, ‎{bf 26} (2010) 95--105‎.]‎. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g...