Displaying similar documents to “ ℵ k -free separable groups with prescribed endomorphism ring”

Prescribing endomorphism algebras of ℵ n -free modules

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2014)

Journal of the European Mathematical Society

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...

Pairs of square-free values of the type n 2 + 1 , n 2 + 2

Stoyan Dimitrov (2021)

Czechoslovak Mathematical Journal

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We show that there exist infinitely many consecutive square-free numbers of the form n 2 + 1 , n 2 + 2 . We also establish an asymptotic formula for the number of such square-free pairs when n does not exceed given sufficiently large positive number.

Separable ℵ k -free modules with almost trivial dual

Daniel Herden, Héctor Gabriel Salazar Pedroza (2016)

Commentationes Mathematicae Universitatis Carolinae

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An R -module M has an almost trivial dual if there are no epimorphisms from M to the free R -module of countable infinite rank R ( ω ) . For every natural number k > 1 , we construct arbitrarily large separable ℵ k -free R -modules with almost trivial dual by means of Shelah’s Easy Black Box, which is a combinatorial principle provable in ZFC.

Unimodular rows over Laurent polynomial rings

Abdessalem Mnif, Morou Amidou (2022)

Czechoslovak Mathematical Journal

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We prove that for any ring 𝐑 of Krull dimension not greater than 1 and n ≥ 3 , the group E n ( 𝐑 [ X , X - 1 ] ) acts transitively on Um n ( 𝐑 [ X , X - 1 ] ) . In particular, we obtain that for any ring 𝐑 with Krull dimension not greater than 1, all finitely generated stably free modules over 𝐑 [ X , X - 1 ] are free. All the obtained results are proved constructively.

Self-small products of abelian groups

Josef Dvořák, Jan Žemlička (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let A and B be two abelian groups. The group A is called B -small if the covariant functor Hom ( A , - ) commutes with all direct sums B ( κ ) and A is self-small provided it is A -small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao, Yuanlin Li, Pingping Zhao, Jujuan Zhuang (2016)

Colloquium Mathematicae

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Let G be an additive finite abelian group. For every positive integer ℓ, let d i s c ℓ ( G ) be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine d i s c ℓ ( G ) for certain finite groups, including cyclic groups, the groups G = C ₂ ⊕ C 2 m and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum...

Equalizers and coactions of groups

Martin Arkowitz, Mauricio Gutierrez (2002)

Fundamenta Mathematicae

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If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group f ⊆ G * H be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism p f = p | f : f → G . A right inverse (section) G → f of p f is called a coaction on G. In this paper we study f and the sections of p f . We consider the following topics: the structure of f as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and...

Consecutive square-free values of the type x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1

Ya-Fang Feng (2023)

Czechoslovak Mathematical Journal

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We show that for any given integer k there exist infinitely many consecutive square-free numbers of the type x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1 . We also establish an asymptotic formula for 1 ≤ x , y , z ≤ H such that x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1 are square-free. The method we used in this paper is due to Tolev.

Large free subgroups of automorphism groups of ultrahomogeneous spaces

Szymon Głąb, Filip Strobin (2015)

Colloquium Mathematicae

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We consider the following notion of largeness for subgroups of S ∞ . A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of S ∞ can be extended to a large free subgroup of S ∞ , and, under Martin’s Axiom, any free subgroup of S ∞ of cardinality less than can also be extended to a large free subgroup of S ∞ . Finally, if Gₙ are countable...

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G , then for every subset B of G with | B | > | G | / k 1 / 3 we have B 3 = G . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if k ≥ 2 , then G has a...

The range of non-linear natural polynomials cannot be context-free

Dömötör Pálvölgyi (2020)

Kybernetika

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Suppose that some polynomial f with rational coefficients takes only natural values at natural numbers, i. e., L = { f ( n ) ∣ n ∈ ℕ } ⊆ ℕ . We show that the base- q representation of L is a context-free language if and only if f is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma.

Number of solutions in a box of a linear equation in an Abelian group

Maciej Zakarczemny (2016)

Colloquium Mathematicae

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For every finite Abelian group Γ and for all g , a ₁ , . . . , a k ∈ Γ , if there exists a solution of the equation ∑ i = 1 k a i x i = g in non-negative integers x i ≤ b i , where b i are positive integers, then the number of such solutions is estimated from below in the best possible way.

Endomorphism kernel property for finite groups

Heghine Ghumashyan, Jaroslav Guričan (2022)

Mathematica Bohemica

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A group G has the endomorphism kernel property (EKP) if every congruence relation θ on G is the kernel of an endomorphism on G . In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.

Limits of relatively hyperbolic groups and Lyndon’s completions

Olga Kharlampovich, Alexei Myasnikov (2012)

Journal of the European Mathematical Society

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We describe finitely generated groups H universally equivalent (with constants from G in the language) to a given torsion-free relatively hyperbolic group G with free abelian parabolics. It turns out that, as in the free group case, the group H embeds into the Lyndon’s completion G ℤ [ t ] of the group G , or, equivalently, H embeds into a group obtained from G by finitely many extensions of centralizers. Conversely, every subgroup of G ℤ [ t ] containing G is universally equivalent to G . Since finitely...

Formally self-referential propositions for cut free classical analysis and related systems

G. Kreisel, G. Takeuti

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CONTENTSIntroduction............................................................................................................................................................................................................ 5  I. Results on self-referential propositions............................................................................................................................. 11    1. Definitions of some principal metamathematical notions......................................................................

A note on certain partial sum operators

Marek Bożejko, Gero Fendler (2006)

Banach Center Publications

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We show that for the t-deformed semicircle measure, where 1/2 < t ≤ 1, the expansions of L p functions with respect to the associated orthonormal polynomials converge in norm when 3/2 < p < 3 and do not converge when 1 ≤ p < 3/2 or 3 < p. From this we conclude that natural expansions in the non-commutative L p spaces of free group factors and of free commutation relations do not converge for 1 ≤ p < 3/2 or 3 < p.

Completely bounded lacunary sets for compact non-abelian groups

Kathryn Hare, Parasar Mohanty (2015)

Studia Mathematica

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In this paper, we introduce and study the notion of completely bounded Λ p sets ( Λ p c b for short) for compact, non-abelian groups G. We characterize Λ p c b sets in terms of completely bounded L p ( G ) multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are Λ p sets for all p < ∞, but are not Λ p c b for any p ≥ 4. This is done by showing that the space of completely bounded L p ( G ) ...

On k -free numbers over Beatty sequences

Wei Zhang (2023)

Czechoslovak Mathematical Journal

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We consider k -free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number α > 1 of finite type τ < ∞ and any constant ε > 0 , we can show that ∑ 1 ≤ n ≤ x [ α n + β ] ∈ 𝒬 k 1 - x ζ ( k ) ≪ x k / ( 2 k - 1 ) + ε + x 1 - 1 / ( τ + 1 ) + ε , where 𝒬 k is the set of positive k -free integers and the implied constant depends only on α , ε , k and β . This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type ∑ 1 ≤ h ≤ H ∑ 1 ≤ n ≤ x n ∈ 𝒬 k e ( ϑ h n ) .

The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group G and a positive real number x , define d G ( x ) to be the number of integers less than x which are dimensions of irreducible complex representations of G . We study the asymptotics of d G ( x ) for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups G in characteristic zero, showing that either there exists α > 0 such that d G ( x ) > x α for all large x , or G is virtually abelian (in which case d G ( x ) is bounded). ...

On a translation property of positive definite functions

Lars Omlor, Michael Leinert (2010)

Banach Center Publications

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If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant C h > 0 such that ∫ L x h · g ≤ C h ∫ h g for every continuous positive definite g≥0, where L x is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at...