ULC Properties in Neighbourhoods of Embedded Surfaces and Curves in E3

1973 ◽  
Vol 25 (1) ◽  
pp. 31-73 ◽  
Author(s):  
J. W. Cannon
Keyword(s):  
Normal Subgroup ◽  
Approximation Theorem ◽  
Sphere Theorem ◽  
Curves And Surfaces ◽  
Local Homology ◽  
Loop Theorem ◽  
Embedded Surfaces ◽  
Nontrivial Normal Subgroup

In this paper we derive those properties of topologically embedded curves and surfaces in E3 which can be obtained without use of Bing's Side Approximation Theorem [3] for surfaces. The local homology and homotopy properties studied classically play the largest role in the paper, but the final geometrization of some of the results requires theorems such as the PL Schoenflies Theorem, Dehn's Lemma, the Loop Theorem, the Sphere Theorem, and Waldhausen's generalization of the Loop Theorem (n.b., one application of Waldhausen's theorem (in (3.4)) requires use of the nontrivial normal subgroup in the statement of that theorem).

scholarly journals Bounds on finite quasiprimitive permutation groups

2001 ◽  
Vol 71 (2) ◽  
pp. 243-258 ◽  
Author(s):  
Cheryl E. Praeger ◽  
Aner Shalev
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Minimum Degree ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Base Size ◽  
Nontrivial Normal Subgroup

AbstractA permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p-elements. We also pose some problems for further research.

Free Groups and Folding

2017 ◽  
Author(s):  
Matt Clay
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Membership Problem ◽  
Research Projects ◽  
Residually Finite ◽  
Simple Operation ◽  
Nontrivial Normal Subgroup ◽  
Subgroup Theorem

This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show the rank of the free group H and whether every finitely generated nontrivial normal subgroup of a free group has finite index. The edge paths and the fundamental group of a graph are discussed, along with subgroups via graphs. The chapter also considers five applications of folding: the Nielsen–Schreier Subgroup theorem, the membership problem, index, normality, and residual finiteness. A group G is residually finite if for every nontrivial element g of G there is a normal subgroup N of finite index in G so that g is not in N. Exercises and research projects are included.

scholarly journals Finite groups with a self-centralizing subgroup of order 4

1967 ◽  
Vol 7 (4) ◽  
pp. 570-576 ◽  
Author(s):  
Warren J. Wong
Keyword(s):  
Normal Subgroup ◽  
Early Result ◽  
Finite Groups ◽  
Present Author ◽  
Permutation Groups ◽  
Odd Order ◽  
Nontrivial Normal Subgroup

The class of finite groups having a subgroup of order 4 which is its own centralizer has been studied by Suzuki [9], Gorenstein and Walter [6], and the present author [11]. The main purpose of this paper is to strengthen Theorem 5 of [11] by using an early result of Zassenhaus [12]. In particular, we find all groups of the class which are core-free, i.e. which have no nontrivial normal subgroup of odd order. As an application, we make a determination of a certain class of primitive permutation groups.

THE NUMBER OF PROFINITE GROUPS WITH A SPECIFIED SYLOW SUBGROUP

2015 ◽  
Vol 99 (1) ◽  
pp. 108-127
Author(s):  
COLIN D. REID
Keyword(s):  
Normal Subgroup ◽  
Sylow Subgroup ◽  
The Other ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Other Hand ◽  
Formula Group ◽  
Nontrivial Normal Subgroup

Let $S$ be a finitely generated pro-$p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$. In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$. For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

A criterion for the existence of nilpotent hall π-subgroups

2020 ◽  
pp. 2250043
Author(s):  
Neda Ahanjideh
Keyword(s):  
Normal Subgroup ◽  
Nontrivial Normal Subgroup ◽  
Text Element

Let [Formula: see text] be a nontrivial normal subgroup of [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] and for every [Formula: see text]-element [Formula: see text] with [Formula: see text], [Formula: see text], for some integer [Formula: see text], then [Formula: see text] has nilpotent Hall [Formula: see text]-subgroups. Further, we show that if [Formula: see text], then [Formula: see text] has abelian Hall [Formula: see text]-subgroups.

Canonical Submersion and Lie Homomorphisms Normal Subgroup

2020 ◽  
Vol 23 (1) ◽  
pp. 97-101
Author(s):  
Mikhail Petrichenko ◽  
Dmitry W. Serow
Keyword(s):  
Normal Subgroup ◽  
Canonical System ◽  
System Of Equations ◽  
Core Group ◽  
The Core ◽  
Switch Group

Normal subgroup module f (module over the ring F = [ f ] 1; 2-diffeomorphisms) coincides with the kernel Ker Lf derivations along the field. The core consists of the trivial homomorphism (integrals of the system v = x = f (t; x )) and bundles with zero switch group Lf , obtained from the condition ᐁ( ω × f ) = 0. There is the analog of the Liouville for trivial immersion. In this case, the core group Lf derivations along the field replenished elements V ( z ), such that ᐁz = ω × f. Hence, the core group Lf updated elements helicoid (spiral) bundles, in particular, such that f = ᐁU. System as an example Crocco shown that the canonical system does not permit the trivial embedding: the canonical system of equations are the closure of the class of systems that permit a submersion.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

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