A Non-abelian, Non-Sidon, Completely Bounded Set

2020 ◽  
pp. 1-7
Author(s):  
Kathryn E. Hare ◽  
Parasar Mohanty
Keyword(s):  
Abelian Group ◽  
Abelian Subgroup ◽  
Sidon Set ◽  
Bounded Set

Abstract The purpose of this note is to construct an example of a discrete non-abelian group G and a subset E of G, not contained in any abelian subgroup, that is a completely bounded $\Lambda (p)$ set for all $p<\infty ,$ but is neither a Leinert set nor a weak Sidon set.

scholarly journals The Relationship Between ϵ-Kronecker Sets and Sidon Sets

2016 ◽  
Vol 59 (3) ◽  
pp. 521-527 ◽  
Author(s):  
Kathryn Hare ◽  
L. Thomas Ramsey
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Dual Group ◽  
Sidon Sets ◽  
Sidon Set ◽  
Arithmetic Properties ◽  
The Fourier Transform ◽  
The Relationship ◽  
Discrete Abelian Group

AbstractA subset E of a discrete abelian group is called ϵ-Kronecker if all E-functions of modulus one can be approximated to within ϵ by characters. E is called a Sidon set if all bounded E-functions can be interpolated by the Fourier transform of measures on the dual group. As ϵ-Kronecker sets with ϵ < 2 possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.

A Lebesgue Decomposition for Vector Valued Additive Set Functions Defined on a Lattice

1977 ◽  
Vol 29 (2) ◽  
pp. 295-298 ◽  
Author(s):  
Thomas P. Dence
Keyword(s):  
Abelian Group ◽  
Projection Operators ◽  
Additive Functions ◽  
Lebesgue Decomposition ◽  
Set Functions ◽  
Bounded Set ◽  
Vector Valued

Our aim is to establish the Lebesgue decomposition for s-bounded vector valued additive functions defined on lattices of sets in both the finitely and countably additive setting. Strongly bounded (s-bounded) set functions were first studied by Rickart [15], and then by Rao [14], Brooks [1] and Darst [5; 6]. In 1963 Darst [6] established a result giving the decomposition of s-bounded elements in a normed Abelian group with respect to an algebra of projection operators.

scholarly journals Fixed points of abelian actions on S2

2007 ◽  
Vol 27 (5) ◽  
pp. 1557-1581 ◽  
Author(s):  
JOHN FRANKS ◽  
MICHAEL HANDEL ◽  
KAMLESH PARWANI
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Abelian Subgroup ◽  
Compact Set ◽  
Finitely Generated

AbstractWe prove that if ${\mathcal F}$ is a finitely generated abelian group of orientation preserving C1 diffeomorphisms of $\mathbb {R}^2$ which leaves invariant a compact set then there is a common fixed point for all elements of ${\mathcal F}$. We also show that if ${\mathcal F}$ is any abelian subgroup of orientation preserving C1 diffeomorphisms of S2 then there is a common fixed point for all elements of a subgroup of ${\mathcal F}$ with index at most two.

scholarly journals CENTRALIZERS OF p-SUBGROUPS IN SIMPLE LOCALLY FINITE GROUPS

2019 ◽  
Vol 62 (1) ◽  
pp. 183-186
Author(s):  
KIVANÇ ERSOY
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Fixed Points ◽  
Simple Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Locally Finite Group ◽  
Simple Groups ◽  
Locally Finite ◽  
Locally Finite Groups

AbstractIn Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra481 (2017), 1–11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikov and for every non-identity α ∈ A the set of fixed points CG(α) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) is Chernikov and for every non-identity α ∈ P the set of fixed points CG(α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ≅PSLp(k) where char k ≠ p and P has a subgroup Q of order p2 such that CG(P) = Q.

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