scholarly journals Groups where non-normal subgroups are close to Abelian groups

2021 ◽  
Vol 19 ◽  
pp. 113
Author(s):  
M.M. Semko ◽  
O.A. Yarova
Keyword(s):  
Commutator Subgroup ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Normal Subgroups

In this paper we consider solvable groups, each subgroup of which is either normal or has a Chernikov commutator subgroup.

EFFICIENT QUANTUM ALGORITHMS FOR SOME INSTANCES OF THE NON-ABELIAN HIDDEN SUBGROUP PROBLEM

2003 ◽  
Vol 14 (05) ◽  
pp. 723-739 ◽  
Author(s):  
GÁBOR IVANYOS ◽  
FRÉDÉRIC MAGNIEZ ◽  
MIKLOS SANTHA
Keyword(s):  
Commutator Subgroup ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Solvable Groups ◽  
Normal Subgroups ◽  
Hidden Subgroup Problem ◽  
Special Cases ◽  
Subgroup Problem ◽  
Small Index ◽  
Hidden Subgroups

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

On normal subgroups of M-groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950074
Author(s):  
Xuewu Chang
Keyword(s):  
Normal Subgroup ◽  
Solvable Group ◽  
Solvable Groups ◽  
Hall Subgroup ◽  
Embedding Theorem ◽  
The Other ◽  
Finite Solvable Group ◽  
Embedding Problem ◽  
Normal Subgroups ◽  
Other Hand

The normal embedding problem of finite solvable groups into [Formula: see text]-groups was studied. It was proved that for a finite solvable group [Formula: see text], if [Formula: see text] has a special normal nilpotent Hall subgroup, then [Formula: see text] cannot be a normal subgroup of any [Formula: see text]-group; on the other hand, if [Formula: see text] has a maximal normal subgroup which is an [Formula: see text]-group, then [Formula: see text] can occur as a normal subgroup of an [Formula: see text]-group under some suitable conditions. The results generalize the normal embedding theorem on solvable minimal non-[Formula: see text]-groups to arbitrary [Formula: see text]-groups due to van der Waall, and also cover the famous counterexample given by Dade and van der Waall independently to the Dornhoff’s conjecture which states that normal subgroups of arbitrary [Formula: see text]-groups must be [Formula: see text]-groups.

THE NUMBER OF CONJUGACY CLASSES CONTAINED IN NORMAL SUBGROUPS OF SOLVABLE GROUPS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250204
Author(s):  
AMIN SAEIDI ◽  
SEIRAN ZANDI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group and let N be a normal subgroup of G. Assume that N is the union of ξ(N) distinct conjugacy classes of G. In this paper, we classify solvable groups G in which the set [Formula: see text] has at most three elements. We also compute the set [Formula: see text] in most cases.

The intersection graph of a group

2015 ◽  
Vol 14 (05) ◽  
pp. 1550065 ◽  
Author(s):  
S. Akbari ◽  
F. Heydari ◽  
M. Maghasedi
Keyword(s):  
Disjoint Union ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Intersection Graph ◽  
Free Graph ◽  
Intersection Graphs ◽  
Cyclic Groups ◽  
Vertex Set

Let G be a group. The intersection graph of G, denoted by Γ(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ 1. In this paper, we show that the girth of Γ(G) is contained in the set {3, ∞}. We characterize all solvable groups whose intersection graphs are triangle-free. Moreover, we show that if G is finite and Γ(G) is triangle-free, then G is solvable. Also, we prove that if Γ(G) is a triangle-free graph, then it is a disjoint union of some stars. Among other results, we classify all abelian groups whose intersection graphs are complete. Finally, we study the intersection graphs of cyclic groups.

scholarly journals On almost finitely generated nilpotent groups

1996 ◽  
Vol 19 (3) ◽  
pp. 539-544 ◽  
Author(s):  
Peter Hilton ◽  
Robert Militello
Keyword(s):  
Nilpotent Group ◽  
Commutator Subgroup ◽  
Local Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Finitely Generated

A nilpotent groupGis fgp ifGp, is finitely generated (fg) as ap-local group for all primesp; it is fg-like if there exists a nilpotent fg groupHsuch thatGp≃Hpfor all primesp. The fgp nilpotent groups form a (generalized) Serre class; the fg-like nilpotent groups do not. However, for abelian groups, a subgroup of an fg-like group is fg-like, and an extension of an fg-like group by an fg-like group is fg-like. These properties persist for nilpotent groups with finite commutator subgroup, but fail in general.

GROUPS WHOSE NONNORMAL SUBGROUPS ARE METAHAMILTONIAN

2019 ◽  
Vol 102 (1) ◽  
pp. 96-103
Author(s):  
DARIO ESPOSITO ◽  
FRANCESCO DE GIOVANNI ◽  
MARCO TROMBETTI
Keyword(s):  
Positive Integer ◽  
Commutator Subgroup ◽  
Abelian Groups

If $\mathfrak{X}$ is a class of groups, we define a sequence $\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots ,\mathfrak{X}_{k},\ldots$ of group classes by putting $\mathfrak{X}_{1}=\mathfrak{X}$ and choosing $\mathfrak{X}_{k+1}$ as the class of all groups whose nonnormal subgroups belong to $\mathfrak{X}_{k}$. In particular, if $\mathfrak{A}$ is the class of abelian groups, $\mathfrak{A}_{2}$ is the class of metahamiltonian groups, that is, groups whose nonnormal subgroups are abelian. The aim of this paper is to study the structure of $\mathfrak{X}_{k}$-groups, with special emphasis on the case $\mathfrak{X}=\mathfrak{A}$. Among other results, it will be proved that a group has a finite commutator subgroup if and only if it is locally graded and belongs to $\mathfrak{A}_{k}$ for some positive integer $k$.

Two-step solvable groups with weak minimal condition for normal subgroups

1986 ◽  
Vol 37 (3) ◽  
pp. 232-236
Author(s):  
L. A. Kurdachenko ◽  
A. V. Tushev
Keyword(s):  
Solvable Groups ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Weak Minimal Condition

scholarly journals Height-zero characters and normal subgroups in p -solvable groups

2014 ◽  
Vol 399 ◽  
pp. 703-711
Author(s):  
A. Laradji
Keyword(s):  
Solvable Groups ◽  
Normal Subgroups

Groups with few conjugacy classes of non-normal subgroups

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Maria Falco ◽  
Francesco Giovanni ◽  
Carmela Musella
Keyword(s):  
Abelian Subgroup ◽  
Soluble Group ◽  
Commutator Subgroup ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
Abelian Subgroups ◽  
Locally Soluble Group

AbstractThe structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.

scholarly journals Some notes on Lie ideals in division rings

2018 ◽  
Vol 17 (03) ◽  
pp. 1850049
Author(s):  
M. Aaghabali ◽  
M. Ariannejad ◽  
A. Madadi
Keyword(s):  
Division Ring ◽  
Commutator Subgroup ◽  
Product Formula ◽  
Main Concern ◽  
Normal Subgroups ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Finitely Generated ◽  
Finite Dimensional ◽  
Lie Product Formula

A Lie ideal of a division ring [Formula: see text] is an additive subgroup [Formula: see text] of [Formula: see text] such that the Lie product [Formula: see text] of any two elements [Formula: see text] is in [Formula: see text] or [Formula: see text]. The main concern of this paper is to present some properties of Lie ideals of [Formula: see text] which may be interpreted as being dual to known properties of normal subgroups of [Formula: see text]. In particular, we prove that if [Formula: see text] is a finite-dimensional division algebra with center [Formula: see text] and [Formula: see text], then any finitely generated [Formula: see text]-module Lie ideal of [Formula: see text] is central. We also show that the additive commutator subgroup [Formula: see text] of [Formula: see text] is not a finitely generated [Formula: see text]-module. Some other results about maximal additive subgroups of [Formula: see text] and [Formula: see text] are also presented.

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