scholarly journals On the permutability of Sylow subgroups with derived subgroups of B-subgroups

2019 ◽  
pp. 12-17
Author(s):  
Ekaterina V. Zubei
Keyword(s):  
Nilpotent Group ◽  
Quotient Group ◽  
Proper Subgroup ◽  
Sylow Subgroups ◽  
Even Order

A finite non-nilpotent group G is called a B-group if every proper subgroup of the quotient group G/Φ(G) is nilpotent. We establish the r-solvability of the group in which some Sylow r-subgroup permutes with the derived subgroups of 2-nilpotent (or 2-closed) B-subgroups of even order and the solvability of the group in which the derived subgroups of 2-closed and 2-nilpotent B-subgroups of even order are permutable.

scholarly journals Nonabelian Sylow subgroups of finite groups of even order

1998 ◽  
Vol 4 (12) ◽  
pp. 88-90
Author(s):  
Naoki Chigira ◽  
Nobuo Iiyori ◽  
Hiroyoshi Yamaki
Keyword(s):  
Finite Groups ◽  
Sylow Subgroups ◽  
Even Order

On permutable strongly 2-maximal and strongly 3-maximal subgroups

2021 ◽  
pp. 121-129
Author(s):  
Yuliya V. Gorbatova
Keyword(s):  
Nilpotent Group ◽  
Solvable Group ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Maximal Subgroups ◽  
Finite Solvable Group ◽  
Prime Divisors ◽  
Sylow Subgroups ◽  
Schmidt Group ◽  
Number Of Prime Divisors

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

Non-abelian Sylow subgroups of finite groups of even order

2000 ◽  
Vol 139 (3) ◽  
pp. 525-539 ◽  
Author(s):  
Naoki Chigira ◽  
Nobuo Iiyori ◽  
Hiroyoshi Yamaki
Keyword(s):  
Finite Groups ◽  
Sylow Subgroups ◽  
Even Order

scholarly journals Some Notes on Relative Commutators

2020 ◽  
Vol 2 (2) ◽  
pp. 65-70
Author(s):  
Masoumeh Ganjali ◽  
Ahmad Erfanian
Keyword(s):  
Nilpotent Group ◽  
Cyclic Group ◽  
Commutator Subgroup ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Perfect Group ◽  
Group A ◽  
Even Order ◽  
Definition Of

Let G be a group and α ϵ Aut(G).  An α-commutator of elements x, y ϵ G is defined as [x, y]α = x-1y-1xyα. In 2015, Barzegar et al. introduced an α-commutator of elements of G and defined a new generalization of nilpotent groups by using the definition of α-commutators which is called an α-nilpotent group. They also introduced an α-commutator subgroup of G, denoted by Dα(G) which is a subgroup generated by all α-commutators. In 2016, an α-perfect group, a group that is equal to its α-commutator subgroup, was introduced by authors of this paper and the properties of such group was investigated. They proved some results on α-perfect abelian groups and showed that a cyclic group G of even order is not α-perfect for any α ϵ Aut(G). In this paper, we may continue our investigation on α-perfect groups and in addition to studying the relative perfectness of some classes of finite p-groups, we provide an example of a non-abelian α-perfect 2-group.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

A note on Lie nilpotent group rings

1992 ◽  
Vol 45 (3) ◽  
pp. 503-506 ◽  
Author(s):  
R.K. Sharma ◽  
Vikas Bist
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Group Rings ◽  
Group Of Units

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

scholarly journals Even-order differential equation with continuous delay: nonexistence criteria of Kneser solutions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ali Muhib ◽  
M. Motawi Khashan ◽  
Osama Moaaz
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Delay Argument ◽  
Even Order ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this paper, we study even-order DEs where we deduce new conditions for nonexistence Kneser solutions for this type of DEs. Based on the nonexistence criteria of Kneser solutions, we establish the criteria for oscillation that take into account the effect of the delay argument, where to our knowledge all the previous results neglected the effect of the delay argument, so our results improve the previous results. The effectiveness of our new criteria is illustrated by examples.

Sign in / Sign up

Export Citation Format

Share Document