scholarly journals On ℱ-residuals of finite groups

2002 ◽  
Vol 65 (2) ◽  
pp. 271-275
Author(s):  
Wenbin Guo ◽  
K. P. Shum ◽  
A. N. Skiba
Keyword(s):  
Finite Groups ◽  
Negative Answer ◽  
Open Problems ◽  
Subdirect Products ◽  
Closed Formation

We prove that there exists a soluble, saturated and s-closed formation ℱ of groups such that the class (Gℱ | G is a group, Gℱ is the ℱ-residual of G) is not closed under subdirect products. This result a negative answer to an open problems recently proposed by L.A. Shemetkov in 1998.

scholarly journals Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside

2019 ◽  
Vol 168 (3) ◽  
pp. 613-633 ◽  
Author(s):  
MARK WILDON
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Open Problems ◽  
Related Literature ◽  
Cyclic Groups ◽  
Computational Data ◽  
Regular Subgroup ◽  
History Of ◽  
Order State

AbstractA group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later, in 1921, he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this paper we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.

CONJUNCTIVE GRAMMARS GENERATE NON-REGULAR UNARY LANGUAGES

2008 ◽  
Vol 19 (03) ◽  
pp. 597-615 ◽  
Author(s):  
ARTUR JEŻ
Keyword(s):  
Negative Answer ◽  
The Body ◽  
Regular Languages ◽  
Natural Numbers ◽  
Open Problems ◽  
Conjunctive Grammars ◽  
Unary Languages ◽  
Context Free ◽  
Conjunctive Grammar ◽  
Context Free Grammars

Conjunctive grammars, introduced by Okhotin, extend context-free grammars by an additional operation of intersection in the body of any production of the grammar. Several theorems and algorithms for context-free grammars generalize to the conjunctive case. Okhotin posed nine open problems concerning those grammars. One of them was a question, whether a conjunctive grammars over a unary alphabet generate only regular languages. We give a negative answer, contrary to the conjectured positive one, by constructing a conjunctive grammar for the language {a4n : n ∈ ℕ}. We also generalize this result: for every set of natural numbers L we show that {an : n ∈ L} is a conjunctive unary language, whenever the set of representations in base-k system of elements of L is regular, for arbitrary k.

On two open problems of the theory of permutable subgroups of finite groups

2019 ◽  
Vol 94 (3-4) ◽  
pp. 477-491
Author(s):  
Bin Hu ◽  
Jianhong Huang ◽  
Alexander N. Skiba
Keyword(s):  
Finite Groups ◽  
Open Problems ◽  
Permutable Subgroups

scholarly journals Global and local properties of finite groups with only finitely many central units in their integral group ring

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andreas Bächle ◽  
Mauricio Caicedo ◽  
Eric Jespers ◽  
Sugandha Maheshwary
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Galois Groups ◽  
Group Rings ◽  
Integral Group ◽  
Open Problems ◽  
Integral Group Rings ◽  
Local Properties ◽  
Global And Local ◽  
The Impact

Abstract The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group, we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same; in particular, the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.

Subdirect products of finite groups with Hall π-subgroups

1996 ◽  
Vol 59 (2) ◽  
pp. 219-221 ◽  
Author(s):  
V. A. Vedernikov
Keyword(s):  
Finite Groups ◽  
Subdirect Products

Finite groups in which every cyclic subgroup is self-normalizing in its subnormal closure

2019 ◽  
Vol 22 (5) ◽  
pp. 941-951
Author(s):  
Guohua Qian
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Formula Group ◽  
A Finite Group ◽  
Closed Formation

Abstract For a given prime p, a finite group G is said to be a {\widetilde{\mathcal{C}}_{p}} -group if every cyclic p-subgroup of G is self-normalizing in its subnormal closure. In this paper, we get some descriptions of {\widetilde{\mathcal{C}}_{p}} -groups, show that the class of {\widetilde{\mathcal{C}}_{p}} -groups is a subgroup-closed formation and that {O^{p^{\prime}}(G)} is a solvable p-nilpotent group for every {\widetilde{\mathcal{C}}_{p}} -group G. We also prove that if a finite group G is a {\widetilde{\mathcal{C}}_{p}} -group for all primes p, then every subgroup of G is self-normalizing in its subnormal closure.

scholarly journals Characteristic classes of involutions in nonsolvable groups

2019 ◽  
Vol 22 (6) ◽  
pp. 1077-1087
Author(s):  
Yotam Fine
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Finite Groups ◽  
Negative Answer ◽  
Characteristic Classes ◽  
Cyclic Subgroups ◽  
Kourovka Notebook ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms ◽  
Made In

Abstract Let {G,D_{0},D_{1}} be finite groups such that {D_{0}\trianglelefteq D_{1}} are groups of automorphisms of G that contain the inner automorphisms of G. Assume that {D_{1}/D_{0}} has a normal 2-complement and that {D_{1}} acts fixed-point-freely on the set of {D_{0}} -conjugacy classes of involutions of G (i.e., {C_{D_{1}}(a)D_{0}<D_{1}} for every involution {a\in G} ). We prove that G is solvable. We also construct a nonsolvable finite group that possesses no characteristic conjugacy class of nontrivial cyclic subgroups. This shows that an assumption on the structure of {D_{1}/D_{0}} above must be made in order to guarantee the solvability of G and also yields a negative answer to Problem 3.51 in the Kourovka notebook, posed by A. I. Saksonov in 1969.

scholarly journals On maximal subformations of n-multiple Ω-foliated formations of finite groups

2021 ◽  
Vol 21 (1) ◽  
pp. 15-25
Author(s):  
Marina M. Sorokina ◽  
◽  
Seraphim P. Maksakov ◽  
Keyword(s):  
Positive Integer ◽  
General Theory ◽  
Finite Groups ◽  
Central Place ◽  
Class I ◽  
Simple Groups ◽  
Subdirect Products ◽  
Maximal Chains

Only finite groups are considered in the article. Among the classes of groups the central place is occupied by classes closed regarding homomorphic images and subdirect products which are called formations. We study Ω-foliateded formations constructed by V. A. Vedernikov in 1999 where Ω is a nonempty subclass of the class I of all simple groups. Ω-Foliated formations are defined by two functions — an Ω-satellite f : Ω ∪ {Ω 0} → {formations} and a direction ϕ : I → {nonempty Fitting formations}. The conception of multiple locality introduced by A. N. Skiba in 1987 for formations and further developed for many other classes of groups, as applied to Ω-foliated formations is as follows: every formation is considered to be 0-multiple Ω-foliated with a direction ϕ; an Ω-foliated formation with a direction ϕ is called an n-multiple Ω-foliated formation where n is a positive integer if it has such an Ω-satellite all nonempty values of which are (n − 1)-multiple Ω-foliated formations with the direction ϕ. The aim of this work is to study the properties of maximal n-multiple Ω-foliated subformations of a given n-multiple Ω-foliated formation. We use classical methods of the theory of groups, of the theory of classes of groups, as well as methods of the general theory of lattices. In the paper we have established the existence of maximal n-multiple Ω-foliated subformations for the formations with certain properties, we have obtained the characterization of the formation ΦnΩϕ (F) which is the intersection of all maximal n-multiple Ω-foliated subformations of the formation F, and we have revealed the relation between a maximal inner Ω-satellite of 1-multiple Ω-foliated formation and a maximal inner Ω-satellite of its maximal 1-multiple Ω-foliated subformation. The results will be useful in studying the inner structure of formations of finite groups, in particular, in studying the maximal chains of subformations and in establishing the lattice properties of formations.

A note on solubly saturated formations of finite groups

2015 ◽  
Vol 14 (04) ◽  
pp. 1550047
Author(s):  
A. Ballester-Bolinches ◽  
S. F. Kamornikov
Keyword(s):  
Finite Groups ◽  
Saturated Formations ◽  
Closed Formation

The main aim of this note is to give a criterion for a subgroup-closed formation to be solubly saturated, which we hope may provide a useful proving ground for outstanding questions about this family of formations.

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