OBTAINING NUCLEI FROM CHAINS OF NORMAL SUBGROUPS

2006 ◽  
Vol 05 (02) ◽  
pp. 215-229 ◽  
Author(s):  
MARK L. LEWIS
Keyword(s):  
Normal Subgroup ◽  
Irreducible Character ◽  
Normal Subgroups ◽  
Separable Group ◽  
Key Concepts ◽  
A Chain

In this paper, we reexamine the foundation of Isaacs' π-theory. One of the key concepts in Isaacs' π-theory is the construction of the characters Bπ(G) for a π-separable group G. The key to determining which characters lie in Bπ(G) was the construction of a nucleus for each irreducible character χ. In this paper, we present a different way of finding a nucleus for χ which is based on a chain of normal subgroup [Formula: see text]. Using this nucleus, we obtain the set of characters [Formula: see text]. We investigate the properties that [Formula: see text] has in common with Bπ(G).

scholarly journals Blocks and Normal Subgroups of Finite Groups

1963 ◽  
Vol 22 ◽  
pp. 15-32 ◽  
Author(s):  
W. F. Reynolds
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Irreducible Character ◽  
Normal Subgroups ◽  
Irreducible Characters ◽  
Section 8 ◽  
A Finite Group

Let H be a normal subgroup of a finite group G, and let ζ be an (absolutely) irreducible character of H. In [7], Clifford studied the irreducible characters X of G whose restrictions to H contain ζ as a constituent. First he reduced this question to the same question in the so-called inertial subgroup S of ζ in G, and secondly he described the situation in S in terms of certain projective characters of S/H. In section 8 of [10], Mackey generalized these results to the situation where all the characters concerned are projective.

scholarly journals Normal subgroups of diffeomorphism and homeomorphism groups of ℝn and other open manifolds

2011 ◽  
Vol 31 (6) ◽  
pp. 1835-1847 ◽  
Author(s):  
PAUL A. SCHWEITZER, S. J.
Keyword(s):  
Normal Subgroup ◽  
Compact Manifold ◽  
Normal Subgroups ◽  
Open Manifold ◽  
Open Manifolds ◽  
Group Of Homeomorphisms ◽  
Homeomorphism Groups

AbstractWe determine all the normal subgroups of the group of Cr diffeomorphisms of ℝn, 1≤r≤∞, except when r=n+1 or n=4, and also of the group of homeomorphisms of ℝn ( r=0). We also study the group A0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with non-empty boundary, then the quotient of A0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

scholarly journals On groups, whose non-normal subgroups are either contranormal or core-free

2020 ◽  
pp. 3-8
Author(s):  
L.A. Kurdachenko ◽  
◽  
A.A. Pypka ◽  
I.Ya. Subbotin ◽  
◽  
...  
Keyword(s):  
Normal Subgroup ◽  
Normal Subgroups

We investigate the influence of some natural types of subgroups on the structure of groups. A subgroup H of a group G is called contranormal in G, if G = HG. A subgroup H of a group G is called core-free in G, if CoreG(H) =〈1〉. We study the groups, in which every non-normal subgroup is either contranormal or core-free. In particular, we obtain the structure of some monolithic and non-monolithic groups with this property

scholarly journals METAPHORIC REPRESENTATION OF THE CONCEPTS OF CRIME AND INVESTIGATION IN DETECTIVE STORIES

2021 ◽  
pp. 24-35
Author(s):  
Doichyk O.Ya. ◽  
Tomash Ya.Z.
Keyword(s):  
Conceptual Metaphor ◽  
Mental Activity ◽  
Cognitive Mechanisms ◽  
Target Domain ◽  
Conceptual Metaphor Theory ◽  
Metaphor Theory ◽  
Key Concepts ◽  
Basic Frame ◽  
Detective Stories ◽  
A Chain

Purpose. The article dwells upon the range of conceptual metaphors with the target domains CRIME and INVESTIGATION verbalized in Arthur Conan Doyle’s detective stories. The research aims at tracing the cognitive mechanisms of conceptual metaphoric mappings which objectify the key concepts of the detective text: CRIME and INVESTIGATION. The analysis is done on the basis of the theoretical points of cognitive linguistic schools, namely the conceptual metaphor theory. The aim is achieved by completing the following tasks: singling out the key concepts of a detective story and tracing their conceptual correlations; schematic representing the basic frame of CRIMINAL INVESTIGATION; analyzing the cognitive mechanisms behind the metaphoric interpretations of CRIME and INVESTIGATION concepts; and describing metaphoric correlations of the basic frame slots (CRIME, CRIMINAL, DETECTIVE, INVESTIGATION).Methods, applied in the research, include contextual and descriptive analysis, conceptual analysis. The range of metaphors with the target domains CRIME and INVESTIGATION is analyzed according to the conceptual metaphor theory methodology.Results. In the detective stories under study the key concepts CRIME, CRIMINAL, DETECTIVE, and INVESTIGATION are represented by a certain set of metaphoric models. The metaphoric expressions that verbalize the concepts of CRIME and INVESTIGATION reveal their conceptual correlations with the concepts of DETECTIVE і CRIMINAL, which obtain further metaphoric interpretation according to these mappings.Conclusions. The research has revealed that the concepts of CRIME and INVESTIGATION have high capacity to be metaphorically interpreted due to their abstract nature. The target domain CRIME is associated with the following set of source domains: PERFORMANCE, GAME, A TANGLE / A PUZZLE / A CHAIN / A RIDDLE / MYSTERY / A LOCKED DOOR, BUSINESS, OCCUPATION, ENTERTAINMENT, MENTAL ACTIVITY, STORY, and PHENOMENON. The range of source domains which correlate with the target domain INVESTIGATION includes: JOURNEY, ROLEPLAY, HUNTING, CHASE, COMPLETING LINKS TO A CHAIN, MAKING VISIBLE, UNTANGLING, and ENTERTAINING ACTIVITY.Key words: frame, conceptual metaphor, range of metaphor, target domain, metaphoric mapping. Статтю присвячено дослідженню діапазону концептуальних метафор для референтів ЗЛОЧИН / CRIME та РОЗСЛІДУВАННЯ / INVESTIGATION у текстах детективних оповідань А. Конан Дойла. Метою статті є простеження когнітивних механізмів творення концептуальних метафор, що об’єктивують ключові концепти оповідань детективного жанру, – ЗЛОЧИН / CRIME та РОЗСЛІДУВАННЯ / INVESTIGATION. Аналіз здійснюється з опертям на положення провідних шкіл когнітивної лінгвістики, зокрема теорії концептуальної метафори. Реалізація поставленої мети відбувається шляхом виокремлення ключових концептів детективної розповіді та з’ясування їхніх концептуальних зв’язків; схематичного моделювання фрейму ДЕТЕКТИВНЕ РОЗСЛІДУВАННЯ / CRIMINAL INVES-TIGATION; аналізу механізмів творення концептуальних метафор, які об’єктивують концепти CRIME та INVESTIGATION і виявляють концептуальні зв’язки між слотами фрейму (CRIME, CRIMINAL, DETECTIVE, INVESTIGATION).Методи, застосовані в дослідженні, включають контекстуальний і концептуальний аналізи, метод суцільної вибірки, описовий метод. Визначення діапазону метафор концептів CRIME та INVESTIGATION здійснюється відповідно до положень теорії концептуальної метафори.Результати. У досліджуваних оповіданнях концепти CRIME, CRIMINAL, DETECTIVE, INVESTIGATION об’єктивовані певним набором метафоричних моделей. У метафоричних висловах, що об’єктивують концепти CRIME та INVESTIGATION у текстах оповідань, відображено їхні зв’язки з концептами DETECTIVE і CRIMINAL, які також отримують своє метафоричне осмислення в межах цих концептуальних метафор.Висновки. Дослідження показує, що високий ступінь метафоризації концептів CRIME та INVESTI-GATION зумовлений абстрактністю референтів. Концептуальний референт CRIME корелює з доменами PERFORMANCE, GAME, A TANGLE / A PUZZLE / A CHAIN / A RIDDLE / MYSTERY / A LOCKED DOOR, BUSINESS, OCCUPATION, ENTERTAINMENT, MENTAL ACTIVITY, STORY, PHENOMENON. Діапазон корелятивних доменів, які проєктуються на референтний домен INVESTIGATION, включає такі кореляти, як JOURNEY, ROLEPLAY, HUNTING, CHASE, COMPLETING LINKS TO A CHAIN, MAKING VISIBLE, UNTANGLING, ENTERTAINING ACTIVITY.Ключові слова: фрейм, концептуальна метафора, діапазон метафори, домен цілі, метафоричне мапування.

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.

Partial characters with respect to a normal subgroup

1999 ◽  
Vol 66 (1) ◽  
pp. 104-124 ◽  
Author(s):  
Gabriel Navarro ◽  
Lucia Sanus
Keyword(s):  
Normal Subgroup ◽  
Complex Space ◽  
Canonical Basis ◽  
Separable Group

AbstractSuppose that G is a π-separable group. Let N be a normal π1-subgroup of G and let H be a Hall π-subgroup of G. In this paper, we prove that there is a canonical basis of the complex space of the class functions of G which vanish of G-conjugates ofHN. When N = 1 and π is the complement of a prime p, these bases are the projective indecomposable characters and set of irreduciblt Brauer charcters of G.

Elementary amenable groups and 4-manifolds with Euler characteristic 0

1991 ◽  
Vol 50 (1) ◽  
pp. 160-170 ◽  
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Euler Characteristic ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Euler Characteristics ◽  
Amenable Groups ◽  
Amenable Subgroup ◽  
Finite Subgroups ◽  
Torsion Free

AbstractWe extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

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.

scholarly journals Normally supportive sublattices of crystallographic space groups

2020 ◽  
Vol 76 (1) ◽  
pp. 7-23
Author(s):  
Miles A. Clemens ◽  
Branton J. Campbell ◽  
Stephen P. Humphries
Keyword(s):  
Normal Subgroup ◽  
Point Group ◽  
Group Extension ◽  
Normal Subgroups ◽  
Symbolic Form ◽  
Space Group ◽  
Space Groups ◽  
Significant Step ◽  
Crystal Class ◽  
Crystallographic Space Groups

The tabulation of normal subgroups of 3D crystallographic space groups that are themselves 3D crystallographic space groups (csg's) is an ambitious goal, but would have a variety of applications. For convenience, such subgroups are referred to as `csg-normal' while normal subgroups of the crystallographic point group (cpg) of a crystallographic space group are referred to as `cpg-normal'. The point group of a csg-normal subgroup must be a cpg-normal subgroup. The present work takes a significant step towards that goal by tabulating the translational subgroups (a.k.a. sublattices) that are capable of supporting csg-normal subgroups. Two necessary conditions are identified on the relative sublattice basis that must be met in order for the sublattice to support csg-normal subgroups: one depends on the operations of the point group of the space group, while the other depends on the operations of the cpg-normal subgroup. Sublattices that meet these conditions are referred to as `normally supportive'. For each cpg-normal subgroup (excluding the identity subgroup 1) of each of the arithmetic crystal classes of 3D space groups, all of the normally supportive sublattices have been tabulated in symbolic form, such that most of the entries in the table contain one or more integer variables of infinite range; thus it could be more accurately described as a table of the infinite families of normally supportive sublattices. For a given pair of cpg-normal subgroup and normally supportive sublattice, csg-normal subgroups of the space groups of the parent arithmetic crystal class can be constructed via group extension, though in general such a pair does not guarantee the existence of a corresponding csg-normal subgroup.

scholarly journals On extension of characters from normal subgroups

1984 ◽  
Vol 27 (1) ◽  
pp. 7-9 ◽  
Author(s):  
G. Karpilovsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Sufficient Condition ◽  
Normal Subgroups ◽  
Complex Character ◽  
A Finite Group

In what follows, character means irreducible complex character.Let G be a finite group and let % be a character of a normal subgroup N. If χ extends to a character of G then χ is stabilised by G, but the converse is false. The aim of this paper is to prove the following theorem which gives a sufficient condition for χ to be extended to a character of G.

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