On the product of a nilpotent group and a group with non-trivial center

Fuzzy subgroups of the direct product of a generalized quaternion group and a cyclic group of any odd order

Bentea and Tu{a}rnu{a}uceanu~(An. c{S}tiinc{t}. Univ. Al. I.Cuza Iac{s}, Ser. Nouv{a}, Mat., {bf 54(1)} (2008), 209-220)proposed the following problem: Find an explicit formula for thenumber of fuzzy subgroups of a finite hamiltonian group of type$Q_8times mathbb{Z}_n$ where $Q_8$ is the quaternion group oforder $8$ and $n$ is an arbitrary odd integer. In this paper weconsider more general grou...

Non-nilpotent Graph of a Group

We associate a graph NG with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of NG and its induced subgraph on G\nil(G), where nil(G) = {x ∈ G | 〈x, y〉 is nilpotent for all y ∈ G}. For any finite group G, we prove that NG has eithe...

Metabelian Product of a Free Nilpotent Group with a Free Abelian Group

In the variety of all groups, A. I. Mal’cev [5] proved in 1949 that the free product of two residually torsion-free nilpotent groups is again residually torsion-free nilpotent. This paper is motivated by the analogous question in the variety of metabelian groups: can we determine whether free metabelian products of residually torsion-free nilpotent metabelian groups are residually torsion-free ...

On the Nilpotent Multiplier of a Free Product

the past hospitalization and its association with suicide attempts and ideation in patients with mdd and comparison with bmd (depressed type) group

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