The existence of pronormal π-Hall subgroups in E π-groups

2015 ◽  
Vol 56 (3) ◽  
pp. 379-383 ◽  
Author(s):  
E. P. Vdovin ◽  
D. O. Revin
Keyword(s):  
Hall Subgroups

On X-Permutable Subgroups

2012 ◽  
Vol 19 (04) ◽  
pp. 699-706
Author(s):  
Baojun Li ◽  
Zhirang Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Hall Subgroups ◽  
Permutable Subgroups

A subgroup A of a group G is said to be X-permutable with another subgroup B in G, where ∅ ≠ X ⊆ G, if there exists some element x ∈ X such that ABx=BxA. In this paper, the solubility and supersolubility of finite groups are described by X-permutability of the Hall subgroups and their subgroups, in addition, the well known theorem of Schur-Zassenhaus in finite group is generalized.

Variations on results on orders of products in finite groups

2020 ◽  
pp. 1-7
Author(s):  
Juan Martínez ◽  
Alexander Moretó
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Simple Groups ◽  
Prime Power Order ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Element Orders ◽  
Hall Subgroups ◽  
A Finite Group

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

ON SYLOW NORMALIZERS OF FINITE GROUPS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350116 ◽  
Author(s):  
L. S. KAZARIN ◽  
A. MARTÍNEZ-PASTOR ◽  
M. D. PÉREZ-RAMOS
Keyword(s):  
Finite Groups ◽  
Property A ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Hall Subgroups ◽  
Saturated Formations ◽  
The Universe ◽  
Do So

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

scholarly journals Computing Hall subgroups of finite groups

2012 ◽  
Vol 15 ◽  
pp. 205-218
Author(s):  
Bettina Eick ◽  
Alexander Hulpke
Keyword(s):  
Finite Groups ◽  
Conjugacy Classes ◽  
Matrix Group ◽  
Effective Algorithm ◽  
Hall Subgroups ◽  
Fitting Model

AbstractWe describe an effective algorithm to compute a set of representatives for the conjugacy classes of Hall subgroups of a finite permutation or matrix group. Our algorithm uses the general approach of the so-called ‘trivial Fitting model’.

scholarly journals Frattini argument for Hall subgroups

2014 ◽  
Vol 414 ◽  
pp. 95-104 ◽  
Author(s):  
Danila Olegovitch Revin ◽  
Evgeny Petrovitch Vdovin
Keyword(s):  
Hall Subgroups

Pronormality of Hall Subgroups in Their Normal Closure

2018 ◽  
Vol 56 (6) ◽  
pp. 451-457 ◽  
Author(s):  
E. P. Vdovin ◽  
M. N. Nesterov ◽  
D. O. Revin
Keyword(s):  
Normal Closure ◽  
Hall Subgroups

Hall subgroups of finite groups

2006 ◽  
pp. 229-263 ◽  
Author(s):  
Danila Olegovitch Revin ◽  
Evgenii Petrovitch Vdovin
Keyword(s):  
Finite Groups ◽  
Hall Subgroups

On Existence of Hall Subgroups in Finite Groups

2017 ◽  
Vol 24 (01) ◽  
pp. 75-82
Author(s):  
Yufeng Liu ◽  
Wenbin Guo ◽  
A.N. Skiba
Keyword(s):  
Finite Groups ◽  
Hall Subgroups

In this paper, we give some new conditions of the existence of Hall subgroups in non-soluble finite groups, and so the famous Hall theorem and Schur-Zassenhaus theorem are generalized.

scholarly journals Finite groups with biprimary Hall subgroups

2015 ◽  
Vol 443 ◽  
pp. 430-440 ◽  
Author(s):  
Valentin N. Tyutyanov ◽  
Viktoryia N. Kniahina
Keyword(s):  
Finite Groups ◽  
Hall Subgroups

Conjugacy of Hall subgroups

1986 ◽  
Vol 56 (1) ◽  
pp. 81-84
Author(s):  
Fletcher Gross
Keyword(s):  
Hall Subgroups
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