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.

On a Result of A. M. Macbeath on Normal Subgroups of a Fuchsian Group

1970 ◽  
Vol 13 (1) ◽  
pp. 15-16 ◽  
Author(s):  
W. Jonsson
Keyword(s):  
Fixed Point ◽  
Normal Subgroup ◽  
Exact Sequence ◽  
Fuchsian Group ◽  
Hyperbolic Plane ◽  
Orbit Space ◽  
Normal Subgroups ◽  
Lefschetz Fixed Point Formula ◽  
Image Position ◽  
Torsion Free

A. M. Macbeath, in November 1965, communicated the following theorem to me which he proved with the aid of the Lefschetz fixed point formula.Theorem. If Γ is a Fuchsian group and N a torsion free normal subgroup, then the rank of N/[Γ, N] is twice the genus of the orbit space D/Γ where D denotes the hyperbolic plane which Γ acts.This theorem will follow from a consideration of the exact sequence*

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.

The normal and subnormal structure of general linear groups

1972 ◽  
Vol 71 (2) ◽  
pp. 163-177 ◽  
Author(s):  
J. S. Wilson
Keyword(s):  
Normal Subgroup ◽  
Linear Group ◽  
Inverse Image ◽  
General Linear ◽  
Normal Subgroups ◽  
General Linear Groups ◽  
Rings Of Algebraic Integers ◽  
Subnormal Structure ◽  
Subgroup Problem ◽  
Image Position

The problem of locating and classifying the normal subgroups of GLn(R), the general linear group of degree n over a commutative ring R with an identity element, has received considerable attention. The solution when R is a field is well known (of. Dieudonné(5), Artin(1)): unless n is equal to two and R has two or three elements, normal subgroups of GLn(R) either lie in the centre of GLn(R) or contain the special linear group SLn(R). However, if R is not a field, then for each ideal I of R the natural map R → R/I induces a homomorphismand, if 0 < I < R, the kernel of θ1 is a non-central normal subgroup of GLn(R) which does not contain SLn(R). The most that may be expected is that each normal subgroup determines an ideal I of R, in such a way that all normal subgroups determining the same ideal I lie between suitably defined greatest and smallest normal subgroups of GLn(R) corresponding to I. For example, write ZI for the inverse image of the centre of GLn(R/I) under the homomorphism θI, and write KI for the intersection of SLn(R) with the kernel of θI. Then KI ≽ ZI, and the results of Klingenberg(7) and Mennicke(9) show that if n ≥ 3, and if R is either a local ring or the ring of rational integers, then any normal subgroup H of GLn(R) satisfiesfor some uniquely determined ideal I. There are many similar theorems. That the above result breaks down for arbitrary rings, even for large n, follows easily from the negative solution to the congruence subgroup problem for certain rings of algebraic integers (see Bass, Milnor and Serre(3)).

COMPONENTS AND MINIMAL NORMAL SUBGROUPS OF FINITE AND PSEUDOFINITE GROUPS

2019 ◽  
Vol 84 (1) ◽  
pp. 290-300
Author(s):  
JOHN S. WILSON
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Minimal Normal Subgroup ◽  
Normal Subgroups ◽  
First Order ◽  
Order Language ◽  
Image Position ◽  
A Finite Group

AbstractIt is proved that there is a formula$\pi \left( {h,x} \right)$in the first-order language of group theory such that each component and each non-abelian minimal normal subgroup of a finite groupGis definable by$\pi \left( {h,x} \right)$for a suitable elementhofG; in other words, each such subgroup has the form$\left\{ {x|x\pi \left( {h,x} \right)} \right\}$for someh. A number of consequences for infinite models of the theory of finite groups are described.

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.

A note on automorphism groups of symmetric cubic graphs

2020 ◽  
pp. 2250018
Author(s):  
Jicheng Ma
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Simple Group ◽  
Minimal Normal Subgroup ◽  
Automorphism Groups ◽  
Cubic Graphs ◽  
Normal Subgroups ◽  
Interesting Consequence ◽  
Cubic Graph

We study [Formula: see text]-arc-transitive cubic graph [Formula: see text], and give a characterization of minimal normal subgroups of the automorphism group. In particular, each [Formula: see text] with quasi-primitive automorphism group is characterized. An interesting consequence of this characterization states that a non-solvable minimal normal subgroup [Formula: see text] contains at most 2 copies of non-abelian simple group when it acts transitively on arcs, or contains at most 6 copies of non-abelian simple group when it acts regularly on vertices.

Some Remarks on IA Automorphisms of Free Groups

1988 ◽  
Vol 40 (5) ◽  
pp. 1144-1155 ◽  
Author(s):  
J. McCool
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Quotient Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Generating Set ◽  
Automorphisms Of Free Groups ◽  
Identity Automorphism ◽  
Image Position

Let An be the automorphism group of the free group Fn of rank n, and let Kn be the normal subgroup of An consisting of those elements which induce the identity automorphism in the commutator quotient group . The group Kn has been called the group of IA automorphisms of Fn (see e.g. [1]). It was shown by Magnus [7] using earlier work of Nielsen [11] that Kn is finitely generated, with generating set the automorphismsandwhere x1, x2, …, xn, is a chosen basis of Fn.

scholarly journals GRUP FAKTOR YANG DIBANGUN DARI SUBGRUP NORMAL FUZZY

2019 ◽  
Vol 13 (1) ◽  
pp. 1
Author(s):  
Mahfuz Tarmizi ◽  
Saman Abdurrahman
Keyword(s):  
Normal Subgroup ◽  
Quotient Group ◽  
Binary Operation ◽  
Normal Subgroups ◽  
Quotient Groups

A Quotient group is a set which contains coset members and satisfies group definition. These cosets are formed by group and its normal subgroup. A set which contains fuzzy coset members is also called a quotient group. These fuzzy cosets are formed by a group and its fuzzy normal subgroup. The purpose of this research is to explain quotient groups induced by fuzzy normal subgroups and isomorphic between them. This research construct sets which contain fuzzy coset members, define an operation between fuzzy cosets and prove these sets under an operation between fuzzy coset satisfy group definition, and prove theorems relating to qoutient groups and homomorphism. The results of this research are  is a qoutient group induced by a fuzzy normal subgroup, where  is a fuzzy normal subgroup of a group ,  is a fuzzy coset, and the binary operation is “” where  for every . An epimorphism  from a group  to a group  and a fuzzy normal subgroup  of  which is constant on  cause quotient goup  and   are isomorphic.

scholarly journals On a Theorem of Hurwitz

1961 ◽  
Vol 5 (2) ◽  
pp. 90-96 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Infinite Family ◽  
Universal Covering ◽  
Fuchsian Groups ◽  
Triangle Group ◽  
Universal Covering Space ◽  
Family Of Curves ◽  
Finite Factor ◽  
Group 2 ◽  
Image Position ◽  
Hurwitz’S Theorem

By a theorem of Hurwitz [3], an algebraic curve of genus g ≧ 2 cannot have more than 84(g − l) birational self-transformations, or, as we shall call them, automorphisms. The bound is attained for Klein's quarticof genus 3 [4]. In studying the problem whether there are any other curves for which the bound is attained, I was led to consider the universal covering space of the Riemann surface, which, as Siegel observed, relates Hurwitz's theorem to Siegel's own result [7] on the measure of the fundamental region of Fuchsian groups. Any curve with 84(g − 1) automorphisms must be uniformized by a normal subgroup of the triangle group (2, 3, 7), and, by a closer analysis of possible finite factor groups of (2, 3, 7), purely algebraic methods yield an infinite family of curves with the maximum number of automorphisms. This will be shown in a later paper.

scholarly journals On normal complements to sections of finite groups

1975 ◽  
Vol 19 (3) ◽  
pp. 257-262 ◽  
Author(s):  
Everett C. Dade
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Subgroups ◽  
Normal Complement ◽  
Image Position ◽  
A Finite Group

Suppose that H/N is a section of a finite group G, i.e., that H is a subgroup of G and N is a normal subgroup of H. We are interested in the existence of normal subgroups M of G satisfying: Such an M can be called a normal complement to the section H/N in G.

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