scholarly journals On Characteristic Morphisms: In Memoriam Thomas Macfarland Cherry

1969 ◽  
Vol 9 (3-4) ◽  
pp. 478-488 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
In Memoriam

This note is concerned with a translation of some concepts and results about characteristic subgroups of a group into the language of categories. As an example, consider strictly characteristic and hypercharacteristic subgroups of a group: the subgroup H of the group G is called strictly characteristic in G if it admits all ependomorphisms of G; that is all homomorphic mappings of G onto G; and H is called hypercharacteristic2 in G if it is the least normal subgroup with factor group isomorphic to G/H, that is if H is contained in every normal subgroup K of G with G/K ≅ G/H.

scholarly journals On some classes of sublattices of the subgroup lattice

2019 ◽  
pp. 35-47 ◽  
Author(s):  
Alexander N. Skiba
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
Subgroup Lattice ◽  
Normal Closure ◽  
Normal Subgroups ◽  
Normal Section ◽  
Normal Core

In this paper G always denotes a group. If K and H are subgroups of G, where K is a normal subgroup of H, then the factor group of H by K is called a section of G. Such a section is called normal, if K and H are normal subgroups of G, and trivial, if K and H are equal. We call any set S of normal sections of G a stratification of G, if S contains every trivial normal section of G, and we say that a stratification S of G is G-closed, if S contains every such a normal section of G, which is G-isomorphic to some normal section of G belonging S. Now let S be any G-closed stratification of G, and let L be the set of all subgroups A of G such that the factor group of V by W, where V is the normal closure of A in G and W is the normal core of A in G, belongs to S. In this paper we describe the conditions on S under which the set L is a sublattice of the lattice of all subgroups of G and we also discuss some applications of this sublattice in the theory of generalized finite T-groups.

scholarly journals Classical groups over division rings of characteristic two

1972 ◽  
Vol 7 (2) ◽  
pp. 191-226 ◽  
Author(s):  
William M. Pender ◽  
G.E. Wall
Keyword(s):  
Normal Subgroup ◽  
Quadratic Form ◽  
Division Ring ◽  
Quadratic Forms ◽  
Factor Group ◽  
Classical Groups ◽  
Isometry Groups ◽  
Division Rings ◽  
Characteristic Two

The notion of quadratic form over a field of characteristic two is extended to an arbitrary division ring of characteristic two with an involution of the first kind. The resulting isometry groups are shown to have a simple normal subgroup and the structure of the factor group is calculated. It is indicated how one may define and analyse all the classical groups in a unified manner by means of quadratic forms.

scholarly journals On supplements in finite groups

1963 ◽  
Vol 3 (1) ◽  
pp. 63-67
Author(s):  
R. Kochendörffer
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Factor Group ◽  
Extension Theory ◽  
A Finite Group

Let G be a finite group. If N denotes a normal subgroup of G, a subgroup S of G is called a supplement of N if we have G = SN. For every normal subgroup of G there is always the trivial supplement S = G. The existence of a non-trivial supplement is important for the extension theory, i.e., for the description of G by means of N and the factor group G/N. Generally, a supplement S is the more useful the smaller the intersection S ∩ N. If we have even S ∩ N = 1, then S is called a complement for N in G. In this case G is a splitting extension of N by S.

scholarly journals FUZZY RINGS AND ITS PROPERTIES

2017 ◽  
Vol 5 (1) ◽  
pp. 32
Author(s):  
Karyati Karyati ◽  
Rifki Chandra Utama
Keyword(s):  
Normal Subgroup ◽  
Fuzzy Sets ◽  
Algebraic Structure ◽  
Binary Operation ◽  
Quotient Ring ◽  
Ring Homomorphism ◽  
Factor Group ◽  
Semi Group ◽  
Fuzzy Ideal ◽  
Fuzzy Ring

Abstract One of algebraic structure that involves a binary operation is a group that is defined  an un empty set (classical) with an associative binary operation, it has identity elements and each element has an inverse. In the structure of the group known as the term subgroup, normal subgroup, subgroup and factor group homomorphism and its properties. Classical algebraic structure is developed to algebraic structure fuzzy by the researchers as an example semi group fuzzy and fuzzy group after fuzzy sets is introduced by L. A. Zadeh at 1965. It is inspired of writing about semi group fuzzy and group of fuzzy, a research on the algebraic structure of the ring is held with reviewing ring fuzzy, ideal ring fuzzy, homomorphism ring fuzzy and quotient ring fuzzy with its properties. The results of this study are obtained fuzzy properties of the ring, ring ideal properties fuzzy, properties of fuzzy ring homomorphism and properties of fuzzy quotient ring by utilizing a subset of a subset level  and strong level  as well as image and pre-image homomorphism fuzzy ring. Keywords: fuzzy ring, subset level, homomorphism fuzzy ring, fuzzy quotient ring

scholarly journals Some infinite Fibonacci groups

1975 ◽  
Vol 19 (3) ◽  
pp. 311-314 ◽  
Author(s):  
D. L. Johnson
Keyword(s):  
Normal Subgroup ◽  
Natural Number ◽  
Free Group ◽  
Factor Group ◽  
Image Position ◽  
Special Case

The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let θ be the automorphism of the free group F = 〈x1, …, xn |〉 of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, …, n). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under θ. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ≦ 2, k ≦ 1,with subscripts reduced modulo n, we obtain the groups F(r, n, k) studied in (1) and (2), while the F(r, n, 1) are the ordinary Fibonacci groups F(r, n) of (3), (5) and (6). To conform with earlier notation, we write A(r, n, k) and A(r, n) for the derived factor groups of F(r, n, k), and F(r, n) respectively.

TTHE CONJUGACY CLASSES OF METABELIAN GROUPS OF ORDER AT MOST 24

2015 ◽  
Vol 77 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Ibrahim Gambo ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Equivalence Relation ◽  
Metabelian Group ◽  
Equivalence Classes ◽  
Factor Group ◽  
Conjugacy Classes ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup

In this paper, G denotes a non-abelian metabelian group and denotes conjugacy class of the element x in G. Conjugacy class is an equivalence relation and it partitions the group into disjoint equivalence classes or sets. Meanwhile, a group is called metabelian if it has an abelian normal subgroup in which the factor group is also abelian. It has been proven by an earlier researcher that there are 25 non-abelian metabelian groups of order less than 24 which are considered in this paper. In this study, the number of conjugacy classes of non-abelian metabelian groups of order less than 24 is computed.

scholarly journals Metanilpotent varieties of groups

2002 ◽  
Vol 73 (1) ◽  
pp. 55-84
Author(s):  
R. M. Bryant ◽  
A. N. Krasil'nikov
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Finite Basis ◽  
Factor Group ◽  
Positive Integers

AbstractFor each positive integer n let N2, n denote the variety of all groups which are nilpotent of class at most 2 and which have exponent dividing n. For positive integers m and n, let N2, mN2, n denote the variety of all groups which have a normal subgroup in N2, m with factor group in N2, n. It is shown that if G ∈N2, mN2, n, where m and n are coprime, then G has a finite basis for its identities.

scholarly journals A Group of Automorphisms of the Homotopy Groups

1951 ◽  
Vol 2 ◽  
pp. 73-82
Author(s):  
Hiroshi Uehara
Keyword(s):  
Normal Subgroup ◽  
Topological Space ◽  
Fundamental Group ◽  
Homotopy Group ◽  
Factor Group ◽  
Complete Analysis ◽  
Homotopy Groups ◽  
Group Of Automorphisms ◽  
Arcwise Connected

It is well known that the fundamental group π1(X) of an arcwise connected topological space X operates on the n-th homotopy group πn(X) of X as a group of automorphisms. In this paper I intend to construct geometrically a group 𝒰(X) of automorphisms of πn(X), for every integer n ≥ 1, which includes a normal subgroup isomorphic to π1(X) so that the factor group of 𝒰(X) by π1(X) is completely determined by some invariant Σ(X) of the space X. The complete analysis of the operation of the group on πn(X) is given in §3, §4, and §5,

Generators for certain normal subgroups of (2,3,7)

1965 ◽  
Vol 61 (2) ◽  
pp. 321-332 ◽  
Author(s):  
John Leech
Keyword(s):  
Normal Subgroup ◽  
Simple Group ◽  
Quotient Group ◽  
Hyperbolic Plane ◽  
Factor Group ◽  
Normal Subgroups ◽  
Finite Factor ◽  
Group 2 ◽  
Image Position ◽  
Symmetry Operations

The infinite groupis the group of direct symmetry operations of the tessellation {3,7} of the hyperbolic plane ((3), chapter 5). This has the smallest fundamental region of any such tessellation, and related to this property is the fact that the group (2, 3, 7) has a remarkable wealth of interesting finite factor groups, corresponding to the finite maps obtained by identifying the results of suitable translations in the hyperbolic plane. The simplest example of this is the group LF(2,7), which is Klein's simple group of order 168. I have studied this group in an earlier paper ((4)), showing inter alia that the group is obtained as a factor group of (2,3,7) by adjoining any one of the relationseach of which implies the others. The method used was to find a set of generators for the normal subgroup with quotient group LF(2,7) and, working entirely within this subgroup, to exhibit that any one of these relations implies its collapse. The technique of working with this subgroup had been developed earlier and applied in (6) to prove that the factor groupis finite and of order 10,752.

scholarly journals Group algebras with an Engel group of units

2006 ◽  
Vol 80 (2) ◽  
pp. 173-178 ◽  
Author(s):  
A. Bovdi
Keyword(s):  
Normal Subgroup ◽  
Group Algebra ◽  
Commutator Subgroup ◽  
Factor Group ◽  
Group Algebras ◽  
Engel Group ◽  
Group Of Units

AbstractLet F be a field of characteristic p and G a group containing at least one element of order p. It is proved that the group of units of the group algebra FG is a bounded Engel group if and only if FG is a bounded Engel algebra, and that this is the case if and only if G is nilpotent and has a normal subgroup H such that both the factor group G/H and the commutator subgroup H′ are finite p–groups.

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