Bases and decomposition numbers of finite groups

Kozma, Gady; Lev, Arieh

doi:10.1007/bf01190111

Cited by 14 publications

(15 citation statements)

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“…Taking h as a real variable, we find that It was also shown by Kozma and Lev [8] that every finite group has a 2-basis A such that |A| ≤ 4 |G|/ √ 3, and so we have the bound…”

Section: /Hmentioning

confidence: 89%

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Rank and Status in Semigroup Theory

Cherubini

Howie

Piochi

2004

Communications in Algebra

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“…Taking h as a real variable, we find that It was also shown by Kozma and Lev [8] that every finite group has a 2-basis A such that |A| ≤ 4 |G|/ √ 3, and so we have the bound…”

Section: /Hmentioning

confidence: 89%

“…• If β moves 1 (with b 1 = 1) and neither β nor γ moves n, then by (8) and (10) we need 3(p − 1) + 3q generators plus 3 generators for the product…”

mentioning

confidence: 99%

Rank and Status in Semigroup Theory

Cherubini

Howie

Piochi

2004

Communications in Algebra

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“…Contributed problems, III: combinatorial and finite geometry (r−1)/2 , where C is an absolute constant, so that any element of F r−1 can be represented as a difference of two elements of D. (The existence of such a subset follows from a general result, proved in [62], and also is not difficult to establish directly.) Now the set D ⊕ {0, 1} determines all directions in F r .…”

Section: Doubling the Squares (Contributed By B Green And T Tao) Hmentioning

confidence: 99%

Open problems in additive combinatorics

Croot¹,

Lev²

2007

CRM Proceedings and Lecture Notes

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Abstract. A brief historical introduction to the subject of additive combinatorics and a list of challenging open problems, most of which are contributed by the leading experts in the area, are presented.In this paper we collect assorted problems in additive combinatorics, including those which we qualify as classical, those contributed by our friends and colleagues, and those raised by the present authors. The paper is organized accordingly: after a historical survey (Section 1) we pass to the classical problems (Section 2), then proceed with the contributed problems (Sections 3-6), and conclude with the original problems (Section 7). Our problem collection is somewhat eclectic and by no means pretends to be complete; the number of problems can be easily doubled or tripled. We tried to include primarily those problems we came across in our research, or at least lying close to the area of our research interests. Additive combinatorics: a brief historical overviewAs the name suggests, additive combinatorics deals with combinatorial properties of algebraic objects, typically abelian groups, rings, or fields. That is, one is interested in those combinatorial properties of the set of elements of an algebraic structure, where the corresponding algebraic operation plays a crucial role. This subject is filled with many wondrous and deep theorems; the earliest of them is, perhaps, the basic CauchyDavenport theorem, proved in 1813 by Cauchy [16] and independently rediscovered in 1935 by Davenport [24,25]. This theorem says that if p is a prime, F p denotes the finite field with p elements (notation used throughout the rest of the paper), and the subsets A, B ⊆ F p are non-empty, then the sumset A + B := {a + b : a ∈ A, b ∈ B} has at least min{p, |A| + |B| − 1} elements. The analogue of this theorem for the set Z of integers is the almost immediate assertion (left as a simple exercise to the interested reader) that |A + B| ≥ |A| + |B| − 1 holds for any finite non-empty subsets A, B ⊆ Z.The F p -version of the problem is considerably more difficult, and all presently known proofs of the Cauchy-Davenport theorem incorporate a non-trivial idea, such as the transform method (sometimes called the "intersection-union trick"), the polynomial method, or Fourier analysis. The situation becomes even more complicated when one 1 2 ERNIE CROOT AND VSEVOLOD F. LEV considers subsets of a general abelian group. An extension of the Cauchy-Davenport theorem onto this case was provided by Kneser whose celebrated result [56,57] asserts that if A and B are finite, non-empty subsets of an abelian group with |A + B| < |A| + |B| − 1, then A + B is a union of cosets of a non-zero subgroup. Further refinement of Kneser's theorem was given by Kemperman in [55].Over a century passed between Cauchy's paper [16] and the next major result in the subject, proved by Schur [83] in the early 1900's. Schur's theorem states that for every fixed integer r > 0 and every r-coloring of the set N of natural numbers, there is a monochromatic triple (x, y, z) ∈ N...

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“…In order to induct on the size of the group, one needs to be able to force some of the elements in a translate of a k-element to be confined to a subset of G of size much smaller than |G| 1−1/k . This prompts the following definition, which is inspired by [11].…”

Section: X|mentioning

confidence: 99%

“…For k = 2 the problem was solved by Kozma and Lev [11] and independently by Finkelstein, Kleitman and Leighton [8] who showed that the easy lower bound above is tight. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%