scholarly journals The efficiency of standard wreath product

2000 ◽  
Vol 43 (2) ◽  
pp. 415-423 ◽  
Author(s):  
A. Sinan Çevik
Keyword(s):  
Wreath Product ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Standard Wreath Product

AbstractLet ξ be the set of all finite groups that have efficient presentations. In this paper we give sufficient conditions for the standard wreath product of two ξ-groups to be a ξ-group.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

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.

scholarly journals Quasinormal Fitting classes of finite groups

2019 ◽  
pp. 18-26
Author(s):  
Martsinkevich Anna V.
Keyword(s):  
Natural Number ◽  
Wreath Product ◽  
Cyclic Group ◽  
Finite Groups ◽  
Fitting Class ◽  
Soluble Groups ◽  
Normal Fitting Class ◽  
Fitting Classes ◽  
Local Fitting ◽  
Class F

Let P be the set of all primes, Zn a cyclic group of order n and X wr Zn the regular wreath product of the group X with Zn. A Fitting class F is said to be X-quasinormal (or quasinormal in a class of groups X ) if F ⊆ X, p is a prime, groups G ∈ F and G wr Zp ∈ X, then there exists a natural number m such that G m wr Zp ∈ F. If  X is the class of all soluble groups, then F is normal Fitting class. In this paper we generalize the well-known theorem of Blessenohl and Gaschütz in the theory of normal Fitting classes. It is proved, that the intersection of any set of nontrivial X-quasinormal Fitting classes is a nontrivial X-quasinormal Fitting class. In particular, there exists the smallest nontrivial X-quasinormal Fitting class. We confirm a generalized version of the Lockett conjecture (in particular, the Lockett conjecture) about the structure of a Fitting class for the case of X-quasinormal classes, where X is a local Fitting class of partially soluble groups.

The Influence of the Number of Non-(sub)normal Non-abelian Subgroups on Solvability of Finite Groups

2016 ◽  
Vol 23 (02) ◽  
pp. 325-328
Author(s):  
Jiangtao Shi ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Abelian Subgroups ◽  
A Finite Group

We obtain some sufficient conditions on the number of non-(sub)normal non-abelian subgroups of a finite group to be solvable, which extend a result of Shi and Zhang in 2011.

CONTROL OF FUSION IN FUSION SYSTEMS

2006 ◽  
Vol 05 (06) ◽  
pp. 817-837 ◽  
Author(s):  
RADU STANCU
Keyword(s):  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fusion System ◽  
Fusion Systems ◽  
Necessary And Sufficient

Let P be a p-group and [Formula: see text] a fusion system on P. The aim of this paper is to give necessary and sufficient conditions on a subgroup Q of P for the normalizer [Formula: see text] to be [Formula: see text] itself. This generalizes a result of Gilotti and Serena on finite groups. As an application we find some classes of resistant p-groups, which are p-groups P such that the normalizer [Formula: see text] is equal to [Formula: see text], for any fusion system [Formula: see text] on P.

Generalization of the basis property on finite groups

2020 ◽  
pp. 2150111
Author(s):  
Ibrahim Al-Dayel ◽  
Ahmad Al Khalaf
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Basis Property ◽  
Generating Set ◽  
Equivalent Basis ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Minimal Generating Set

A group [Formula: see text] has the Basis Property if every subgroup [Formula: see text] of [Formula: see text] has an equivalent basis (minimal generating set). We studied a special case of the finite group with the Basis Property, when [Formula: see text]-group [Formula: see text] is an abelian group. We found the necessary and sufficient conditions on an abelian [Formula: see text]-group [Formula: see text] of [Formula: see text] with the Basis Property to be kernel of Frobenius group.

Solubility theorems for finite groups

1973 ◽  
Vol 73 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Thomas J. Laffey
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
A Finite Group

In this paper we obtain various sufficient conditions for the solubility of a finite group. In particular, we show that if G is a finite group and p≥5 is a prime such that all p′-subgroups of G are nilpotent, then G is soluble. We show also that if G is a finite group which has a cyclic Sylow p-subgroup Pand such that for all p′-subgroups H of G, H is nilpotent and H′ is cyclic, then, if p≠3, either P◃G or G has a normal p-complement.

scholarly journals The pseudoidentity problem and reducibility for completely regular semigroups

2001 ◽  
Vol 63 (3) ◽  
pp. 407-433 ◽  
Author(s):  
Jorge Almedia ◽  
Peter G. Trotter
Keyword(s):  
Word Problem ◽  
Finite Groups ◽  
Open Problem ◽  
Sufficient Conditions ◽  
90Th Birthday ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Strengthened Version ◽  
Completely Regular Semigroups ◽  
Necessary And Sufficient

Dedicated to George Szekeres on the occasion of his 90th birthdayNecessary and sufficient conditions for equality over the pseudovariety CR of all finite completely regular semigroups are obtained. They are inspired by the solution of the word problem for free completely regular semigroups and clarify the role played by groups in the structure of such semigroups. A strengthened version of Ash's inevitability theorem (κ-reducibility of the pseudovariety G of all finite groups) is proposed as an open problem and it is shown that, if this stronger version holds, then CR is also κ-reducible and, therefore, hyperdecidable.

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