Finite groups with a given number of relative centralizers

We will study all finite groups and their subgroups with at most four relative centralizers.     

Finite 2-groups all of whose nonabelian subgroups are generated with involutions

In this note we determine finite nonabelian 2-groups G all of whose nonabelian subgroups are generated by involutions and show that such groups must be quasi-dihedral. This solves the problem Nr. 1595 for p = 2 in [1].     

Finite groups with transitive semipermutability

A group G is said to be a T-group (resp. PT-group, PST-group), if normality (resp. permutability, S-permutability) is a transitive relation. In this paper, we get the characterization of finite solvable PST-groups. We also give a new characterization of finite solvable PT-groups.      

Finite groups with complemented minimal p-subgroups

Abstract

For a given prime p, we investigate the finite groups in which every minimal p-subgroup is complemented.     

Finite groups of matrices over group rings

We investigate certain finite subgroups of , where is a finite nilpotent group. Such a group gives rise to a -module; we study the characters of these modules to limit the structure of . We also exhibit some exotic subgroups .

     

Finite groups with Hall supplements to primitive subgroups

Using the classification of finite simple groups, we establish the solvability of every finite group with solvable Hall supplements to primitive subgroups. We prove also that if those supplements are Ore dispersive then so is the group itself.     

Finite groups with S-permutable n-minimal subgroups

We study the supersolvability of finite groups and the nilpotent length of finite solvable groups under the assumption that all their exactly n-minimal subgroups are S-permutable, where n is an arbitrary integer.     

Finite groups with small centralizers of word-values

Given a positive integer m and a group-word w, we consider a finite group G such that $$w(G) ne 1$$ and all centralizers of non-trivial w-values have order at most m. We prove that if $$w=v(x_1^{q_1},dots ,x_k^{q_k})$$, where v is a multilinear commutator word and $$q_1, dots , q_k$$ are p-powers for some prime p, then the order of G is bounded in terms of w and m only. Similar results hold when w is the nth Engel word or the word $$w=[x^n, y_1, dots ,y_k]$$ with $$k ge 1$$.     

Homogeneous monomial groups and centralizers

The construction of homogeneous monomial groups are given and their basic properties are studied. The structure of a centralizer of an element is completely described and the problem of conjugacy of two elements is resolved. Moreover, the classification of homogeneous monomial groups are determined by using the lattice of Steinitz numbers, namely, we prove the following: Let λ and μ be two Steinitz numbers. The homogeneous monomial groups Σ_λ(H) and Σ_μ(G) are isomorphic if and only if λ = μ and H?G provided that the splittings of Σ_λ(H) and Σ_μ(G) are regular.     

Finite p-groups All of Whose Minimal Nonabelian Subgroups are Nonmetacyclic of Order p3

Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p³. Let P₁-groups denote the p-groups all of whose minimal nonabelian subgroups are nonmetacyclic of order p³. In this paper, the P₁-groups are classified, and as a by-product, we prove the Hughes' conjecture is true for the P₁-groups.     

On finite groups with a certain number of centralizers

LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)?A₅, then #Cent(G)=22 or 32. Moroever, we prove that A₅ is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A₅ in terms of the number of centralizers