Pronormality of Hall Subgroups in Their Normal Closure

2018 ◽  
Vol 56 (6) ◽  
pp. 451-457 ◽  
Author(s):  
E. P. Vdovin ◽  
M. N. Nesterov ◽  
D. O. Revin
2007 ◽  
Vol 10 ◽  
pp. 341-353 ◽  
Author(s):  
Michael Vaughan-Lee
Keyword(s):  

In this note, the author proves that a group G is a 4-Engel group if and only if the normal closure of every element g ∈ G is a 3-Engel group


2017 ◽  
Vol 60 (2) ◽  
pp. 333-338 ◽  
Author(s):  
GREGOR MASBAUM

AbstractWe use elementary skein theory to prove a version of a result of Stylianakis (Stylianakis, The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere, arXiv:1511.02912) who showed that under mild restrictions on m and n, the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.


1983 ◽  
Vol 26 (1) ◽  
pp. 89-96 ◽  
Author(s):  
James Howie

Let G be a group, and let r = r(t) be an element of the free product G * 〈G〉 of G with the infinite cyclic group generated by t. We say that the equation r(t) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * 〈G〉 to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to 〈G, t|r〉 (the quotient of G * 〈G〉 by the normal closure of r) being injective.


Author(s):  
ANDRÉ MACEDO ◽  
RACHEL NEWTON

Abstract Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\] . We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group.


2019 ◽  
Vol 11 (02) ◽  
pp. 273-292
Author(s):  
Charalampos Stylianakis

In this paper we show that the normal closure of the [Formula: see text]th power of a half-twist has infinite index in the mapping class group of a punctured sphere if [Formula: see text] is at least five. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the [Formula: see text]th power of a Dehn twist has infinite order, and for some integers [Formula: see text] the quotient contains a free group. As a second corollary we recover a result of Coxeter: the normal closure of the [Formula: see text]th power of a half-twist in the braid group of at least four strands has infinite index. Our method is to reformulate the Jones representation of the mapping class group of a punctured sphere, using the action of Hecke algebras on [Formula: see text]-graphs, as introduced by Kazhdan–Lusztig.


2015 ◽  
Vol 56 (3) ◽  
pp. 379-383 ◽  
Author(s):  
E. P. Vdovin ◽  
D. O. Revin
Keyword(s):  

2012 ◽  
Vol 19 (04) ◽  
pp. 699-706
Author(s):  
Baojun Li ◽  
Zhirang Zhang

A subgroup A of a group G is said to be X-permutable with another subgroup B in G, where ∅ ≠ X ⊆ G, if there exists some element x ∈ X such that ABx=BxA. In this paper, the solubility and supersolubility of finite groups are described by X-permutability of the Hall subgroups and their subgroups, in addition, the well known theorem of Schur-Zassenhaus in finite group is generalized.


1990 ◽  
Vol 108 (3) ◽  
pp. 467-474 ◽  
Author(s):  
John Hempel

For X a subset of a group G, the smallest normal subgroup of G which contains X is called the normal closure of X and is denoted by ngp (X; G) or simply by ngp (X) if there is no possibility of ambiguity. By a surface group we mean the fundamental group of a compact surface. We are interested in determining when a normal subgroup of a surface group contains a simple loop – the homotopy class of an embedding of S1 in the surface, or more generally, a power of a simple loop. This is significant to the study of 3-manifolds since a Heegaard splitting of a 3-manifold is reducible (cf. [2]) if and only if the kernel of the corresponding splitting homomorphism contains a simple loop. We give an answer in the case that the normal subgroup is the normal closure ngp (α) of a single element α: if ngp (α) contains a (power of a) simple loop β then α is homotopic to a (power of a) simple loop and β±1 is homotopic either to (a power of) α or to the commutator [α, γ] of a with some simple loop γ meeting a transversely in a single point. This implies that if a is not homotopic to a power of a simple loop, then the quotient map π1(S) → π1(S)/ngp (α) does not factor through a group with more than one end. In the process we show that π1(S)/ngp (α) is locally indicable if and only if α is not a proper power and that α always lifts to a simple loop in the covering space Sα of S corresponding to ngp (α). We also obtain some estimates on the minimal number of double points in certain homotopy classes of loops.


Development ◽  
1993 ◽  
Vol 117 (3) ◽  
pp. 1163-1172 ◽  
Author(s):  
H.W. van Straaten ◽  
J.W. Hekking ◽  
C. Consten ◽  
A.J. Copp

Neurulation has been suggested to involve both factors intrinsic and extrinsic to the neuroepithelium. In the curly tail (ct) mutant mouse embryo, final closure of the posterior neuropore is delayed to varying extents resulting in neural tube defects. Evidence was presented recently (Brook et al., 1991 Development 113, 671–678) to suggest that enhanced ventral curvature of the caudal region is responsible for the neurulation defect, which probably originates from an abnormally reduced rate of cell proliferation affecting the hindgut endoderm and notochord, but not the neuroepithelium (Copp et al., 1988, Development 104, 285–295). This axial curvature probably generates a mechanical stress on the posterior neuropore, opposing normal closure. We predicted, therefore, that the ct/ct posterior neuropore should be capable of normal closure if the neuropore should be capable of normal closure if the neuroepithelium is isolated from its adjacent tissues. This prediction was tested by in vitro culture of ct/ct posterior neuropore regions, isolated by a cut caudal to the 5th from last somite. In experimental explants, the neuroepithelium of the posterior neuropore, together with the contiguous portion of the neural tube, were separated mechanically from all adjacent non-neural tissues. The posterior neuropore closed in these explants at a similar rate to isolated posterior neuropore regions of non-mutant embryos. By contrast, control ct/ct explants, in which the caudal region was isolated but the neuroepithelium was left attached to adjacent tissues, showed delayed neurulation. To examine further the idea that axial curvature may be a general mechanism regulating neurulation, we cultured chick embryos on curved substrata in vitro. Slight curvature of the body axis (maximally 1 degree per mm axial length), of either concave or convex nature, resulted in delay of posterior neuropore closure in the chick embryo. Both incidence and extent of closure delay correlated with the degree of curvature that was imposed. We propose that during normal embryogenesis the rate of neurulation is related to the angle of axial curvature, such that experimental alterations in curvature will have differing effects (either enhancement or delay of closure) depending on the angle of curvature at which neurulation normally occurs in a given species, or at a given level of the body axis.


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