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.

An inverse theorem for additive bases

2016 ◽  
Vol 12 (06) ◽  
pp. 1509-1518 ◽  
Author(s):  
Yongke Qu ◽  
Dongchun Han
Keyword(s):  
Abelian Group ◽  
Proper Subgroup ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Regular Sequence ◽  
Inverse Theorem ◽  
Additive Basis

Let [Formula: see text] be a finite abelian group of order [Formula: see text], and [Formula: see text] be the smallest prime dividing [Formula: see text]. Let [Formula: see text] be a sequence over [Formula: see text]. We say that [Formula: see text] is regular if for every proper subgroup [Formula: see text], [Formula: see text] contains at most [Formula: see text] terms from [Formula: see text]. Let [Formula: see text] be the smallest integer [Formula: see text] such that every regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] forms an additive basis of [Formula: see text], i.e. [Formula: see text]. Recently, [Formula: see text] was determined for many abelian groups. In this paper, we determined [Formula: see text] for more abelian groups and characterize the structure of the regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] and [Formula: see text].

scholarly journals ADDITIVE BASES IN ABELIAN GROUPS

2010 ◽  
Vol 06 (04) ◽  
pp. 799-809
Author(s):  
VSEVOLOD F. LEV ◽  
MIKHAIL E. MUZYCHUK ◽  
ROM PINCHASI
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Vector Space ◽  
Abelian Groups ◽  
Unit Cube ◽  
Strong Form ◽  
Additive Basis

Let G be a finite, non-trivial Abelian group of exponent m, and suppose that B1,…, Bk are generating subsets of G. We prove that if k > 2m ln log2 |G|, then the multiset union B1 ∪ Bk forms an additive basis of G; that is, for every g ∈ G, there exist A1 ⊆ B1,…, Ak ⊆ Bk such that [Formula: see text]. This generalizes a result of Alon, Linial and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where B1,…, Bk are finite subsets of a vector space, we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form [Formula: see text], where Ai vary over all subsets of Bi for each i = 1,…, k. Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings.

Stable cohomotopy and cobordism of abelian groups

1981 ◽  
Vol 90 (2) ◽  
pp. 273-278 ◽  
Author(s):  
C. T. Stretch
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Formal Group ◽  
Characteristic Class ◽  
Burnside Ring ◽  
Classifying Space ◽  
Formal Group Law ◽  
Stable Cohomotopy ◽  
Group Law

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

EXTREMAL CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 125-135 ◽  
Author(s):  
ABBY GAIL MASK ◽  
JONI SCHNEIDER ◽  
XINGDE JIA
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Communication Networks ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Element Subset ◽  
Cayley Digraph ◽  
Positive Integers ◽  
Element Set

Cayley digraphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m*(d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k-element subset A of Γ such that diam ( Cay (Γ, A)) ≤ d, where diam ( Cay (Γ, A)) denotes the diameter of the Cayley digraph Cay (Γ, A) of Γ generated by A. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam (ℤm, A)) ≤ d. In this paper, we prove, among other results, that [Formula: see text] for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley digraph is optimal with respect to its diameter and degree can be a cyclic group.

scholarly journals EXPLICIT cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf–Witten invariants

2019 ◽  
Vol 150 (4) ◽  
pp. 1937-1964 ◽  
Author(s):  
Hua-Lin Huang ◽  
Zheyan Wan ◽  
Yu Ye
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Bar Resolution ◽  
Chain Map ◽  
A Chain

AbstractWe provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.

scholarly journals The Number of Subgroup Chains of Finite Nilpotent Groups

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1537 ◽  
Author(s):  
Lingling Han ◽  
Xiuyun Guo
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Recursive Formula ◽  
Nilpotent Groups ◽  
Classification Problem ◽  
Finite Abelian Groups ◽  
Counting Problem ◽  
Cyclic Groups ◽  
Fuzzy Subgroups

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

ON A PARTITION PROBLEM OF FINITE ABELIAN GROUPS

2015 ◽  
Vol 92 (1) ◽  
pp. 24-31
Author(s):  
ZHENHUA QU
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Partition Problem ◽  
Finite Abelian Groups ◽  
Disjoint Sets ◽  
Ordered Pairs

Let$G$be a finite abelian group and$A\subseteq G$. For$n\in G$, denote by$r_{A}(n)$the number of ordered pairs$(a_{1},a_{2})\in A^{2}$such that$a_{1}+a_{2}=n$. Among other things, we prove that for any odd number$t\geq 3$, it is not possible to partition$G$into$t$disjoint sets$A_{1},A_{2},\dots ,A_{t}$with$r_{A_{1}}=r_{A_{2}}=\cdots =r_{A_{t}}$.

Relative autocommutator subgroups of abelian groups

2017 ◽  
Vol 16 (05) ◽  
pp. 1750086
Author(s):  
Mohammad Naghshinehfard
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Groups Of Automorphisms

We compute the autocentral and autoderived series of a finite abelian group [Formula: see text] corresponding to several groups of automorphisms [Formula: see text]. As a result, we determine the autonilpotency and autosolvability of [Formula: see text] with respect to [Formula: see text].

Additive bases of Cp ⊕ Cpn

2017 ◽  
Vol 13 (09) ◽  
pp. 2453-2459
Author(s):  
Yongke Qu ◽  
Dongchun Han
Keyword(s):  
Abelian Group ◽  
Proper Subgroup ◽  
Finite Abelian Group ◽  
Regular Sequence ◽  
Additive Basis

Let [Formula: see text] be a finite abelian group, and [Formula: see text] be the smallest prime dividing [Formula: see text]. Let [Formula: see text] be a sequence over [Formula: see text]. We say that [Formula: see text] is regular if for every proper subgroup [Formula: see text], [Formula: see text] contains at most [Formula: see text] terms from [Formula: see text]. Let [Formula: see text] be the smallest integer [Formula: see text] such that every regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] forms an additive basis of [Formula: see text], i.e. [Formula: see text]. In this paper, we show that [Formula: see text] where [Formula: see text].

scholarly journals On the number of subgroups of a given exponent in a finite abelian group

2017 ◽  
Vol 101 (115) ◽  
pp. 121-133 ◽  
Author(s):  
Marius Tărnăuceanu ◽  
László Tóth
Keyword(s):  
Abelian Group ◽  
Asymptotic Formula ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Explicit Formulas

This paper deals with the number of subgroups of a given exponent in a finite abelian group. Explicit formulas are obtained in the case of rank two and rank three abelian groups. An asymptotic formula is also presented.

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