scholarly journals On finite groups whose derived subgroup has bounded rank

2010 ◽  
Vol 178 (1) ◽  
pp. 51-60 ◽  
Author(s):  
K. Podoski ◽  
B. Szegedy
Keyword(s):  
Finite Groups ◽  
Derived Subgroup ◽  
Bounded Rank

AN APPROACH TO CAPABLE GROUPS AND SCHUR’S THEOREM

2015 ◽  
Vol 92 (1) ◽  
pp. 52-56
Author(s):  
MITRA HASSANZADEH ◽  
RASOUL HATAMIAN
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finiteness Condition ◽  
Schur’S Theorem ◽  
Derived Subgroup ◽  
Capable Groups ◽  
A Finite Group ◽  
Bounded Rank

Podoski and Szegedy [‘On finite groups whose derived subgroup has bounded rank’, Israel J. Math.178 (2010), 51–60] proved that for a finite group $G$ with rank $r$, the inequality $[G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$ holds. In this paper we omit the finiteness condition on $G$ and show that groups with finite derived subgroup satisfy the same inequality. We also construct an $n$-capable group which is not $(n+1)$-capable for every $n\in \mathbf{N}$.

Finite groups with two supersoluble subgroups

2019 ◽  
Vol 22 (2) ◽  
pp. 297-312 ◽  
Author(s):  
Victor S. Monakhov ◽  
Alexander A. Trofimuk
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Derived Subgroup ◽  
Nilpotent Residual ◽  
A Finite Group

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

Coprime Group Actions Fixing All Nonlinear Irreducible Characters

1989 ◽  
Vol 41 (1) ◽  
pp. 68-82 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Group Actions ◽  
Character Degrees ◽  
Prime Divisors ◽  
Irreducible Characters ◽  
Nilpotent Length ◽  
Derived Subgroup ◽  
Nonlinear Irreducible Character

The main result of this paper is the following:Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the setcd(G) = ﹛x(l)lx ∈ Irr(G) ﹜of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.

Automorphisms of finite groups of bounded rank

1993 ◽  
Vol 82 (1-3) ◽  
pp. 395-404 ◽  
Author(s):  
Aner Shalev
Keyword(s):  
Finite Groups ◽  
Bounded Rank

scholarly journals Products of Finite Connected Subgroups

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1498
Author(s):  
María Pilar Gállego ◽  
Peter Hauck ◽  
Lev S. Kazarin ◽  
Ana Martínez-Pastor ◽  
María Dolores Pérez-Ramos
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Groups ◽  
Derived Subgroup ◽  
A Finite Group

For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if ⟨a,b⟩∈L for all a∈A and b∈B. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.

A Characterization of the Class of Finite Groups with Nilpotent Derived Subgroup

2002 ◽  
Vol 239-240 (1) ◽  
pp. 5-10
Author(s):  
A. Ballester-Bolinches ◽  
L.M. Ezquerro ◽  
M.C. Pedraza-Aguilera
Keyword(s):  
Finite Groups ◽  
Derived Subgroup

scholarly journals GROUPS IN WHICH NORMAL CLOSURES OF ELEMENTS HAVE BOUNDEDLY FINITE RANK

2009 ◽  
Vol 51 (2) ◽  
pp. 341-345 ◽  
Author(s):  
PATRIZIA LONGOBARDI ◽  
MERCEDE MAJ ◽  
HOWARD SMITH
Keyword(s):  
Finite Rank ◽  
Normal Closure ◽  
Derived Subgroup ◽  
Bounded Rank

AbstractIt is proved that if the normal closure of every element of a group G has rank at most r, then the derived subgroup of G has r-bounded rank.

scholarly journals FINITE GROUPS WITH ENGEL SINKS OF BOUNDED RANK

2018 ◽  
Vol 60 (3) ◽  
pp. 695-701 ◽  
Author(s):  
E. I. KHUKHRO ◽  
P. SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
A Finite Group ◽  
Bounded Rank

AbstractFor an element g of a group G, an Engel sink is a subset ${\mathscr E}$(g) such that for every x ∈ G all sufficiently long commutators [. . .[[x, g], g], . . ., g] belong to ${\mathscr E}$(g). A finite group is nilpotent if and only if every element has a trivial Engel sink. We prove that if in a finite group G every element has an Engel sink generating a subgroup of rank r, then G has a normal subgroup N of rank bounded in terms of r such that G/N is nilpotent.

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