Narrow Results

We characterize the varieties in which all soluble groups are torsion-by-nilpotent as well as the varieties in which all soluble groups are locally finite-by-nilpotent.

Let G be a group, R be a ring and A be an RG-module. We say that A is an Artinian-finitary module overRG if for every element g ∈ G, the factor-module A/C_A(g) is an Artinian R-module. The study of these modules was initiated by Wehrfritz. If D is a Dedekind domain and U is an Artinian D-module, then we can associate with U some numerical invariants. If V is the maximal divisible submodule of U, then V is a direct sum of finitely many indecomposable submodules. The number b_d(U) of these direct summands is an invariant of U. The composition length b_F(U/V) of U/V is another invariant of U. We consider the following special case of Artinian-finitary modules. Let D be a Dedekind domain and G be a group. The DG-module A is said to be bounded Artinian-finitary, if A is Artinian-finitary and there are the numbers b_F(A) = b, b_d(A) = d ∈ ℕ and a finite subset b_σ(A) = τSpec(D) such that l_F(A/C_A(g)) ≤ b, l_d(A/C_A(g)) ≤ d and Ass_D(A/C_A(g)) ⊆ τ for every element g ∈ G. In the article, the bounded Artinian-finitary modules under some natural restriction are studied.

We established several new criteria for existence of complements and supplements to some normal abelian subgroups in groups. In passing, as one of the many useful applications and corollaries of these results, we obtained a description of some finitely generated soluble groups of finite Hirsch–Zaitsev rank. As another application of our results, we obtained a D.J.S. Robinson's theorem on structure of finitely generated soluble groups of finite section rank. The original proof of this theorem was homological, but all proofs in this paper, including this one, are purely group-theoretical.

We study free subgroups in the multiplicative group of division rings generated by group rings of some classes of soluble groups.

In this paper, we study the class (Ω, ∞) of groups whose every infinite subset contains two distinct elements generating an Ω-group where Ω is either the class of Černikov groups, or the class of Černikov-by-nilpotent groups and we deduce some characterizations of finite-by-nilpotent groups.

For a soluble finite group $G$ and a prime $p$ we let $u_p(G)={max}_{H◃◃G}\{j:p^j\top |H:H_G|\}$, $u(G)={max}_{p\in \pi (G)}{u}_p(G)$. We obtain upper bounds for the rank, the nilpotent length, the derived length, and the $p$-length of a finite soluble group $G$ in terms of $u_p(G)$ and $u(G)$.

Gagola and Lewis proved that a finite group $G$ is nilpotent if and only if $\chi {(1)}^2$ divides $|G:$$\mathrm{\text{Ker}}$$\chi |$ for all irreducible characters $\chi$ of $G$. In this paper, we prove that a finite soluble group $G$ is nilpotent if and only if $\chi {(1)}^2$ divides $|G:$$\mathrm{\text{Ker}}$$\chi |$ for all irreducible monomial characters $\chi$ of $G$.

For a group $G$, let $\gamma (G)$ denote the number of conjugacy classes of non-nilpotent subgroups of $G$. We classify all finite groups $G$ with $\gamma (G)\le 3$

The solubility of a finite group with less than $53$ supersoluble subgroups is confirmed in this paper. Moreover, we prove that a finite insoluble group has exactly $53$ supersoluble subgroups if and only if it is isomorphic to $A_5$. Furthermore, it is shown that a finite group with less than $7$ conjugacy classes of supersoluble subgroups is soluble, and the only finite insoluble group with $7$ conjugacy classes of supersoluble subgroups is isomorphic to $A_5$.

Given a class $𝔉$ of finite groups, we consider the graph ${\tilde{\mathrm{\Gamma }}}_{𝔉}(G)$ whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $〈g,h〉\notin 𝔉$. Moreover, we denote by ${{ℐ}}_{𝔉}(G)$ the set of the isolated vertices of ${\tilde{\mathrm{\Gamma }}}_{𝔉}(G)$ and by ${\mathrm{\Gamma }}_{𝔉}(G)$ the graph obtained from ${\tilde{\mathrm{\Gamma }}}_{𝔉}(G)$ by deleting the isolated vertices. We address the following question: to what extent the fact that ${{ℐ}}_{𝔉}(H)$ is a subgroup of $H$ for any $H\le G,$ implies that the graph ${\mathrm{\Gamma }}_{𝔉}(G)$ is connected?

Let G be a finite group. We let and σ (G) denote the number of maximal subgroups of G and the least positive integer n such that G is written as the union of n proper subgroups, respectively. In this paper, we determine the structure of G/ Φ (G) when G is a finite soluble group with .

In this paper, we study some properties of the non-abelian tensor product of two groups G and H. More precisely, if G is abelian and H is a nilpotent group, then an upper bound for the exponent of G ⊗ H is obtained. Using our results, we obtain some upper bounds for the exponent of the Schur multiplier of the non-abelian tensor product of groups. Finally, an abelian group is constructed by taking non-abelian tensor product of groups.

We consider groups G such that N_G(H)/HC_G(H) is cyclic for all H ≤ G. More specifically, we characterise locally nilpotent and supersoluble groups with this property.

We consider finite groups in which every triple of distinct conjugacy class sizes greater than one has a pair which is coprime. We prove such a group is soluble and has conjugate rank at most three.

The importance of conjugacy classes for the structure of finite groups was recognised very early in the study of groups. In this survey we consider the results from the many articles which have developed this topic and examined the influence of conjugacy class sizes or the number of conjugacy classes on the structure of finite groups. Whilst we begin by mentioning the early results of Sylow and Burnside, our major objective is to highlight the more recent work and present some interesting questions which we hope will inspire further research.

We obtained some sufficient and necessary conditions of existence of faithful irreducible representations of a soluble group G of finite rank over a field k. It was shown that the existence of such representations strongly depends on construction of the socle of the group G. The situation is especially complicated in the case where the field k is locally finite.