scholarly journals Quantitative norm convergence of double ergodic averages associated with two commuting group actions

2014 ◽  
Vol 36 (3) ◽  
pp. 860-874 ◽  
Author(s):  
VJEKOSLAV KOVAČ
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Group Actions ◽  
Norm Convergence ◽  
Ergodic Averages ◽  
Direct Sum ◽  
Measure Preserving Actions

We study double averages along orbits for measure-preserving actions of$\mathbb{A}^{{\it\omega}}$, the direct sum of countably many copies of a finite abelian group$\mathbb{A}$. We show an$\text{L}^{p}$norm-variation estimate for these averages, which in particular re-proves their convergence in$\text{L}^{p}$for any finite$p$and for any choice of two$\text{L}^{\infty }$functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.

scholarly journals On the group ring of a finite abelian group

1969 ◽  
Vol 1 (2) ◽  
pp. 245-261 ◽  
Author(s):  
Raymond G. Ayoub ◽  
Christine Ayoub
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Finite Abelian Group ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Group Of Units ◽  
Direct Sum ◽  
A Finite Group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

The corepresentation ring of a pointed Hopf algebra of rank one

2018 ◽  
Vol 17 (12) ◽  
pp. 1850236
Author(s):  
Zhihua Wang
Keyword(s):  
Abelian Group ◽  
Hopf Algebra ◽  
Nilpotent Element ◽  
Finite Abelian Group ◽  
Generators And Relations ◽  
Direct Sum ◽  
Ring Form ◽  
Nilpotent Elements ◽  
Finite Dimensional ◽  
Rank One

Let [Formula: see text] be an arbitrary pointed Hopf algebra of rank one and [Formula: see text] the group of group-like elements of [Formula: see text]. In this paper, we give the decomposition of a tensor product of finite dimensional indecomposable right [Formula: see text]-comodules into a direct sum of indecomposables. This enables us to describe the corepresentation ring of [Formula: see text] in terms of generators and relations. Such a ring is not commutative if [Formula: see text] is not abelian. We describe all nilpotent elements of the corepresentation ring of [Formula: see text] if [Formula: see text] is a finite abelian group or a particular Hamiltonian group. In this case, all nilpotent elements of the corepresentation ring form a principal ideal which is either zero or generated by a nilpotent element of degree 2.

scholarly journals INTERCHANGE RINGS

2016 ◽  
Vol 101 (3) ◽  
pp. 310-334
Author(s):  
CHARLES C. EDMUNDS
Keyword(s):  
Abelian Group ◽  
Prime Power ◽  
Binary Operation ◽  
Finite Abelian Group ◽  
Linear Operators ◽  
Prime Power Order ◽  
Cyclic Groups ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Interchange Law

An interchange ring,$(R,+,\bullet )$, is an abelian group with a second binary operation defined so that the interchange law$(w+x)\bullet (y+z)=(w\bullet y)+(x\bullet z)$ holds. An interchange near ring is the same structure based on a group which may not be abelian. It is shown that each interchange (near) ring based on a group $G$ is formed from a pair of endomorphisms of $G$ whose images commute, and that all interchange (near) rings based on $G$ can be characterized in this manner. To obtain an associative interchange ring, the endomorphisms must be commuting idempotents in the endomorphism semigroup of $G$. For $G$ a finite abelian group, we develop a group-theoretic analogue of the simultaneous diagonalization of idempotent linear operators and show that pairs of endomorphisms which yield associative interchange rings can be diagonalized and then put into a canonical form. A best possible upper bound of $4^{r}$ can be given for the number of distinct isomorphism classes of associative interchange rings based on a finite abelian group $A$ which is a direct sum of $r$ cyclic groups of prime power order. If $A$ is a direct sum of $r$ copies of the same cyclic group of prime power order, we show that there are exactly ${\textstyle \frac{1}{6}}(r+1)(r+2)(r+3)$ distinct isomorphism classes of associative interchange rings based on $A$. Several examples are given and further comments are made about the general theory of interchange rings.

scholarly journals Inverse results for weighted Harborth constants

2016 ◽  
Vol 12 (07) ◽  
pp. 1845-1861 ◽  
Author(s):  
Luz E. Marchan ◽  
Oscar Ordaz ◽  
Dennys Ramos ◽  
Wolfgang A. Schmid
Keyword(s):  
Inverse Problem ◽  
Abelian Group ◽  
Inverse Problems ◽  
Cyclic Group ◽  
Finite Abelian Group ◽  
Maximal Length ◽  
Direct Sum ◽  
Inverse Results

For a finite abelian group [Formula: see text], the Harborth constant is defined as the smallest integer [Formula: see text] such that each squarefree sequence over [Formula: see text] of length [Formula: see text] has a subsequence of length equal to the exponent of [Formula: see text] whose terms sum to [Formula: see text]. The plus-minus weighted Harborth constant is defined in the same way except that the existence of a plus-minus weighted subsum equaling [Formula: see text] is required, that is, when forming the sum one can choose a sign for each term. The inverse problem associated to these constants is the problem of determining the structure of squarefree sequences of maximal length that do not yet have such a zero-subsum. We solve the inverse problems associated to these constants for certain groups, in particular, for groups that are the direct sum of a cyclic group and a group of order two. Moreover, we obtain some results for the plus-minus weighted Erdős–Ginzburg–Ziv constant.

Galois Extensions as Modules over the Group Ring

1970 ◽  
Vol 22 (2) ◽  
pp. 242-248 ◽  
Author(s):  
Gerald Garfinkel ◽  
Morris Orzech
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Galois Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Finite Abelian Group ◽  
Galois Extensions ◽  
Normal Bases ◽  
Direct Sum ◽  
Definition Of

Suppose that R is a commutative ring and G is a finite abelian group. In § 2 we review the definition of E(R, G) (T(R, G)), the group of all (commutative) Galois extensions S of R with Galois group G. We discuss the properties of these groups as functors of G and give an example which exhibits some of the pathological properties of the functor E(R, – ). In § 3 we display a homomorphism from E(R, G) to Pic (R(G)); we use this homomorphism to prove that if S is commutative, G has exponent m, and R(G) has Serre dimension 0 or 1, then a direct sum of m copies of S is isomorphic as a G-module to a direct sum of m copies of R(G). (This result is related to [5, Theorem 4.2], where it is shown that if S is a free R-module and G is any finite group with n elements, then Sn is isomorphic to R(G)n as G-modules.) We also give some examples of Galois extensions without normal bases.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

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