On Preservation of Stability for Finite Extensions of Abelian Groups

We characterize preservation of superstability and ω-stability for finite extensions of abelian groups and reduce the general case to the case of p-groups. In particular we study finite extensions of divisible abelian groups. We prove that superstable abelian-by-finite groups have only finitely many conjugacy classes of Sylow p-subgroups. Mathematics Subject Classification: 03C60, 20C05.     

<Emphasis Type="Italic">p</Emphasis>-groups having few almost-rational irreducible characters

We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ _p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p ² and p ² + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p ² + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p ², and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p ³ are exceptional.     

Schur multiplicators of finite <Emphasis Type="Italic">p</Emphasis>-groups with fixed coclass

We investigate the Schur multiplicators M(G) of p-groups G using coclass theory. For p > 2 we show that there are at most finitely many p-groups G of coclass r with |M(G)| ≤ s for every r and s. We observe that this is not true for p = 2 by constructing infinite series of 2-groups G with coclass r and |M(G)| = 1. We investigate the Schur multiplicators of the 2-groups of coclass r further.     

Nice bases for primary Abelian groups

Abstract Suppose that A is an Abelian p-group. It is proved that if p^ωA is bounded, then A has a bounded nice basis and if p^ωA is a direct sum of cyclic groups, then A has a nice basis. In particular, all Abelian p-groups of length < ω.2 along with all simply presented Abelian p-groups are equipped with bounded nice bases. It is also shown that if length(A)≤ ω.2 and A/p^ωA is countable, then A possesses a bounded nice basis as well as if length(A)≤ ω.2 and p^ωA is countable, then A possesses a nice basis. Moreover, contrasting with these claims, we demonstrate that if length(A)=ω.2 and A/p^ωA is torsion-complete with finite Ulm-Kaplansky invariants, then A does not have a bounded nice basis. If in addition p^ωA is torsion-complete, then A does not have a nice basis, respectively. Finally, we construct a summable -projective group (thus a summable group with a nice basis) which is not a direct sum of countable groups. This answers in negative our question posed in (Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia, 2005). Keywords: Bounded nice basis, Nice basis, Bounded groups, Direct sums of cyclic groups, Summable groups, -projective groups, Simply presented groups, Σ-groups, Torsion-complete groups, Large subgroups, Countable extensions, Bounded extensions Mathematics Subject Classification: 20K10, 20K15 An erratum to this article is available at .     

New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups

A partial difference set having parameters (n ², r(n − 1), n + r ² − 3r, r ² − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n ², r(n + 1), − n + r ² + 3r, r ² + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form where the s _i are nonnegative integers and s ₀ + s ₁ ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form for nonnegative integers s _i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.      

On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

Note on elongations of totally projective p-groups by p ω+n -projective p-groups

It is proved that any Σ-group, which is a special elongation of a totally projective abelian p-group by a p ^ω+1-projective abelian p-group, is totally projective. In particular, each p ^ω+1-projective abelian Σ-p-group is a direct sum of countable p-groups of lengths not exceeding ω + 1. This strengthens our recent result published in Comment. Math. Univ. St. Pauli (2006). Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 180–185, April–June, 2006.     

Primary abelian <Emphasis Type="Italic">n</Emphasis>-Σ-groups

We prove the following theorem: Any abelian p-group is an n-Σ-group which is a strong ω-elongation of a totally projective group by a p ^ω+n -projective group precisely when it is totally projective. In particular, each p-torsion p ^ω+n -projective n-Σ-group is a direct sum of countable p-groups of length not exceeding ω + n and vice versa. These two claims generalize our recent results in [6] and [7]. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 155–162, April–June, 2007.     

A classification of some regular p-groups and its applications

In this paper we classify regular p-groups with type invariants (e/it, 1, 1, 1) for e⩾2 and (1, 1, 1, 1, 1). As a by-product, we give a new approach to the classification of groups of order p ⁵, p⩾5 a prime.     

I λ-Groups and almost totally injective groups

Summary We introduce the class ofI _λ-groups, where λ is an ordinal, consisting of these abelianp-groupsG such thatG/p ^σ G is almost totally injective for every ordinal σ<λ. As an application of some results onI _λ-groups, we prove an extension property of almost totally injective groups.

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Riassunto Si introduce la classe degliI _λ-gruppi, dove λ è un ordinale, formata daip-gruppi abelianiG tali cheG/p ^σ G è quasi totalmente iniettivo per ogni ordinale σ<λ. Come applicazione di alcuni risultati sugliI _λ-gruppi, si prova una proprietà di estensione dei gruppi quasi totalmente iniettivi.

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Lavoro eseguito nell'ambito dei Gruppi di Ricerca Matematica del Consiglio Nazionale delle Ricerche.     

Zero sets and multiplier theorems for star-invariant sub spaces

Given an inner function θ, let {Kskθ/p}:= H^p ∩θ {Hsk0/p} be the corresponding star-invariant subspace of the Hardy spaceH ^p. We show that, unless θ is a finite Blaschke product, the zero sets for K _θ ^p -spaces are different for different p’s. We also investigate the (non)stability of zero sets when passing from {Kskθ/p} to {Ksku/q}, whereq > p and u is an inner function divisible by θ. This problem is motivated by the Beurling-Malliavin multiplier theorem for entire functions, and we solve it (at least in a natural special case) by proving an appropriate multiplier theorem for K _θ ^p .     

Normal complements in modp-envelopes

SupposeG is a finitep-group andk is the field ofp elements, and letU be the augmentation ideal of the group algebrakG. We investigate whichp-groups,G, have normal complements in their modp-envelope,G ^*.G ^* is defined byG ^*={1−u∶u∈U}.     

Embeddings in Lattice-O *-Groups

In the paper, the following results are obtained: the existence of simple divisible lattice-O ^*-groups is established (Theorem 2.1) and it is proved that any countable lattice-orderable or right-orderable group can be isomorphically embedded in a simple divisible lattice-O ^*-group (Theorem 2.2 and Corollary 2.3).     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

A braidlike presentation of Sp(n,p)

Forn even andp an odd prime a symplectic group Sp(n, p) is a quotient of the Artin braid groupB _n+1. Ifs ₁, …,s _n are standard generators ofB _n+1 then the kernel of the corresponding epimorphism is the normal closure of just four elements:s ₁ ^p ,(s ₁ s ₂)⁶,s ₁ ^(p+1)/2 s ₂ ⁴ s ₁ ^(p−1)/2 s ₂ ⁻² s ₁ ⁻¹ s ₂ ² and (s ₁ s ₂ s ₃)⁴ A ⁻¹ s ₁ ⁻² A, whereA=s ₂ s ₃ ⁻¹ s ₂ ^(p−1)/2 s ₄ s ₃ ² s ₄, all of them lying in the subgroupB ₅. Sp(n, p) acts on a vector space and the image of the subgroupB _n ofB _n+1 in Sp(n, p), denoted Sp(n−1,p), is a stabilizer of one vector. A sequence of inclusions …B _k+1·B _k … induces a sequence of inclusions …Sp(k,p)·Sp(k−1,p)…, which can be used to study some finite-valued invariants of knots and links in the 3-sphere via the Markov theorem. Partially supported by the Technion VPR-Fund.     

How many slopes in a polygon?

The compactness theorem for the predicate calculus is used to prove that ifp is a prime ands is a positive integer withs≦11 or 2 ^s−1≦p and X is a set of distinct residues modp, there are at least 2s−3 distinct residuesx+y withx≠y andx, y∈X.     

Kolmogorov and linear widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ∈ ℕ and let Δ ₊ ^s be the set of functions x: I ↦ ℝ on a finite interval I such that the divided differences [x; t ₀, ..., t _s ] of order s of these functions are nonnegative for all collections of s + 1 different points t ₀, ..., t _s ∈ I. For the classes Δ ₊ ^s B _p : = Δ ₊ ^s ∩ B _p , where B _p is the unit ball in L _p , we determine the orders of Kolmogorov and linear widths in the spaces _Lq for 1 ≤ q > p ≤ ∞. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1633–1652, December, 2005.     

Sets of rigged paths with Virasoro characters

Let {M _r,s ^(p,p′)}_{1≤r≤p−1,1≤s≤p′−1} be the irreducible Virasoro modules in the (p,p′)-minimal series. In our previous paper, we have constructed a monomial basis of ⊕ _r=1 ^p−1 M _r,s ^(p,p′) in the case 1<p′/p<2. By ‘monomials’ we mean vectors of the form , where φ _−n ^(r′,r):M _r,s ^(p,p′)→M _r′,s ^(p,p′) are the Fourier components of the (2,1)-primary field and |r ₀,s〉 is the highest weight vector of . In this article, we introduce for all p<p′ with p≥3 and s=1 a subset of such monomials as a conjectural basis of ⊕ _r=1 ^p−1 M _r,1^(p,p′). We prove that the character of the combinatorial set labeling these monomials coincides with the character of the corresponding Virasoro module. We also verify the conjecture in the case p=3.