scholarly journals On growth of generalized Grigorchuk's overgroups

2020 ◽  
Vol 30 (1) ◽  
pp. 97-117
Author(s):  
S. T. Samarakoon ◽  
Keyword(s):  
Infinite Order ◽  
Polynomial Growth ◽  
Torsion Group ◽  
Intermediate Growth ◽  
Branch Group ◽  
The Family

Grigorchuk's Overgroup G˜, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group G of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of G. The group G, corresponding to the sequence (012)∞=012012…, is a member of the family {Gω|ω∈Ω={0,1,2}N} consisting of groups of intermediate growth when sequence ω is not eventually constant. Following this construction we define the family {G˜ω,ω∈Ω} of generalized overgroups. Then G˜=G˜(012)∞ and Gω is a subgroup of G˜ω for each ω∈Ω. We prove, if ω is eventually constant, then G˜ω is of polynomial growth and if ω is not eventually constant, then G˜ω is of intermediate growth.

scholarly journals G-Hypergroups: Hypergroups with a Group-Isomorphic Heart

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 240
Author(s):  
Mario De Salvo ◽  
Dario Fasino ◽  
Domenico Freni ◽  
Giovanni Lo Faro
Keyword(s):  
Group Actions ◽  
Torsion Group ◽  
Large Classes ◽  
Trivial Group ◽  
The Family ◽  
Nontrivial Group ◽  
The Right

Hypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is the trivial group. However, very little is known about hypergroups that are neither 1-hypergroups nor belong to the first class. The goal of this work is to take a first step in classifying G-hypergroups, that is, hypergroups whose heart is a nontrivial group. We introduce their main properties, with an emphasis on G-hypergroups whose the heart is a torsion group. We analyze the main properties of the stabilizers of group actions of the heart, which play an important role in the construction of multiplicative tables of G-hypergroups. Based on these results, we characterize the G-hypergroups that are of type U on the right or cogroups on the right. Finally, we present the hyperproduct tables of all G-hypergroups of size not larger than 5, apart of isomorphisms.

scholarly journals The distribution of prime ideals of a Dedekind domain

1974 ◽  
Vol 11 (3) ◽  
pp. 429-441 ◽  
Author(s):  
Anne P. Grams
Keyword(s):  
Class Group ◽  
Sufficient Conditions ◽  
Torsion Group ◽  
Dedekind Domain ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Class Groups ◽  
Maximal Ideals ◽  
The Family ◽  
Necessary And Sufficient

Let G be an abelian group, and let S be a subset of G. Necessary and sufficient conditions on G and S are given in order that there should exist a Dedekind domain D with class group G with the property that S is the set of classes that contain maximal ideals of D. If G is a torsion group, then S is the set of classes containing the maximal ideals of D if and only if S generates G. These results are used to determine necessary and sufficient conditions on a family {Hλ} of subgroups of G in order that there should exist a Dedekind domain D with class group G such that {G/Hλ} is the family of class groups of the set of overrings of D. Several applications are given.

scholarly journals Continuous Cocycle Superrigidity for the Full Shift Over a Finitely Generated Torsion Group

2018 ◽  
Vol 2020 (6) ◽  
pp. 1610-1620 ◽  
Author(s):  
David Bruce Cohen
Keyword(s):  
Discrete Group ◽  
Infinite Order ◽  
Torsion Group ◽  
Finitely Generated ◽  
Full Shift ◽  
Order Element

Abstract Chung and Jiang showed that if a one-ended group contains an infinite order element, then every continuous cocycle over the full shift, taking values in a discrete group, must be cohomologous to a homomorphism. We show that their conclusion holds for all one-ended groups, so that the hypothesis of admitting an infinite order element may be omitted.

THE BURNSIDE PROBLEM FOR GROUPS OF LOW QUADRATIC GROWTH

1996 ◽  
Vol 06 (03) ◽  
pp. 369-377
Author(s):  
ROBERTO INCITTI
Keyword(s):  
Infinite Order ◽  
Polynomial Growth ◽  
Growth Function ◽  
Infinite Group ◽  
Quadratic Growth ◽  
Burnside Problem ◽  
Finitely Generated ◽  
Combinatorial Argument ◽  
Groups Of Polynomial Growth ◽  
Generating System

We show with a combinatorial argument that a finitely generated infinite group whose growth function relative to some finite generating system is less or equal to [Formula: see text], r<2, contains an element of infinite order. This result is aimed at investigating the combinatorial nature of M. Gromov’s theorem on groups of polynomial growth.

Generalized Grigorchuk’s overgroups as points in the space of marked 8-generated groups

2020 ◽  
pp. 2250058
Author(s):  
Supun T. Samarakoon
Keyword(s):  
Grigorchuk Group ◽  
Intermediate Growth ◽  
The Family ◽  
Self Similar

First Grigorchuk group [Formula: see text] and Grigorchuk’s overgroup [Formula: see text], introduced in 1980, are self-similar branch groups with intermediate growth. In 1984, [Formula: see text] was used to construct the family of generalized Grigorchuk groups [Formula: see text], which has many remarkable properties. Following this construction, we generalize the Grigorchuk’s overgroup [Formula: see text] to the family [Formula: see text] of generalized Grigorchuk’s overgroups. We consider these groups as 8-generated and describe the closure of this family in the space [Formula: see text] of marked [Formula: see text]-generated groups.

Triassic sponges (Sphinctozoa) from Hells Canyon, Oregon

1988 ◽  
Vol 62 (03) ◽  
pp. 419-423 ◽  
Author(s):  
Baba Senowbari-Daryan ◽  
George D. Stanley
Keyword(s):  
Endemic Species ◽  
Upper Triassic ◽  
Island Arc ◽  
Tropical Island ◽  
The Family ◽  
Wallowa Terrane ◽  
Unique Condition ◽  
Hells Canyon

Two Upper Triassic sphinctozoan sponges of the family Sebargasiidae were recovered from silicified residues collected in Hells Canyon, Oregon. These sponges areAmblysiphonellacf.A. steinmanni(Haas), known from the Tethys region, andColospongia whalenin. sp., an endemic species. The latter sponge was placed in the superfamily Porata by Seilacher (1962). The presence of well-preserved cribrate plates in this sponge, in addition to pores of the chamber walls, is a unique condition never before reported in any porate sphinctozoans. Aporate counterparts known primarily from the Triassic Alps have similar cribrate plates but lack the pores in the chamber walls. The sponges from Hells Canyon are associated with abundant bivalves and corals of marked Tethyan affinities and come from a displaced terrane known as the Wallowa Terrane. It was a tropical island arc, suspected to have paleogeographic relationships with Wrangellia; however, these sponges have not yet been found in any other Cordilleran terrane.

Electron Microscopy of Mycoplasma Pneumoniae

1971 ◽  
Vol 29 ◽  
pp. 258-259 ◽  
Author(s):  
E. S. Boatman ◽  
G. E. Kenny
Keyword(s):  
Electron Microscopy ◽  
Light Microscopy ◽  
Growth Phase ◽  
Hela Cells ◽  
Sulfonic Acid ◽  
Gamma Globulin ◽  
Horse Serum ◽  
Morphological Aspects ◽  
The Family

Information concerning the morphology and replication of organism of the family Mycoplasmataceae remains, despite over 70 years of study, highly controversial. Due to their small size observations by light microscopy have not been rewarding. Furthermore, not only are these organisms extremely pleomorphic but their morphology also changes according to growth phase. This study deals with the morphological aspects of M. pneumoniae strain 3546 in relation to growth, interaction with HeLa cells and possible mechanisms of replication.The organisms were grown aerobically at 37°C in a soy peptone yeast dialysate medium supplemented with 12% gamma-globulin free horse serum. The medium was buffered at pH 7.3 with TES [N-tris (hyroxymethyl) methyl-2-aminoethane sulfonic acid] at 10mM concentration. The inoculum, an actively growing culture, was filtered through a 0.5 μm polycarbonate “nuclepore” filter to prevent transfer of all but the smallest aggregates. Growth was assessed at specific periods by colony counts and 800 ml samples of organisms were fixed in situ with 2.5% glutaraldehyde for 3 hrs. at 4°C. Washed cells for sectioning were post-fixed in 0.8% OSO4 in veronal-acetate buffer pH 6.1 for 1 hr. at 21°C. HeLa cells were infected with a filtered inoculum of M. pneumoniae and incubated for 9 days in Leighton tubes with coverslips. The cells were then removed and processed for electron microscopy.

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