Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

Lower bound on the profile of degree pairs in cross-intersecting set systems

We raise the following problem. For natural numbers m, n ≥ 2, determine pairs d′, d″ (both depending on m and n only) with the property that in every pair of set systems A, B with |A| ≤ m, |B| ≤ n, and A ∩ B ≠ 0 for all A ∈ A, B ∈ B, there exists an element contained in at least d′ |A| members of A and d″ |B| members of B. Generalizing a previous result of Kyureghyan, we prove that all the extremal pairs of (d′, d″) lie on or above the line (n − 1) x + (m − 1) y = 1. Constructions show that the pair (1 + ɛ / 2n − 2, 1 + ɛ / 2m − 2) is infeasible in general, for all m, n ≥ 2 and all ɛ > 0. Moreover, for m = 2, the pair (d′, d″) = (1 / n, 1 / 2) is feasible if and only if 2 ≤ n ≤ 4. The problem originates from Razborov and Vereshchagin’s work on decision tree complexity. Research supported in part by the Hungarian Research Fund under grant OTKA T-032969.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

On extending Pollard’s theorem for <Emphasis Type="Italic">t</Emphasis>-representable sums

Let t ≥ 1, let A and B be finite, nonempty subsets of an abelian group G, and let $ A\mathop + \limits_i B $ A\mathop + \limits_i B denote all the elements c with at least i representations of the form c = a + b, with a ∈ A and b ∈ B. For |A|, |B| ≥ t, we show that either

On the number of popular differences

We prove that there exists an absolute constant c > 0 such that for any finite set A ⊆ ℤ with |A| ≥ 2 and any positive integer m < c|A|/ ln |A| there are at most m integers b > 0 satisfying |(A + b) \ A| ≤ m; equivalently, there are at most m positive integers possessing |A| −m (or more) representations as a difference of two elements of A.     

On a combinatorial problem in group theory

We say that a groupG ∈DS if for some integerm, all subsetsX ofG of sizem satisfy |X ²|<|X|², whereX ²={xy|x,y ∈X}. It is shown, using a previous result of Peter Neumann, thatG ∈DS if and only if either the subgroup ofG generated by the squares of elements ofG is finite, orG contains a normal abelian subgroup of finite index, on which each element ofG acts by conjugation either as the identity automorphism or as the inverting automorphism. Dedicated to John G. Thompson, the Wolf Prize Laureate in Mathematics for 1992 The first author wishes to thank the Department of Mathematics in the University of Napoli for their hospitality during the preparation of this paper.     

When do random subsets decompose a finite group?

Let A, B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group, i.e. [AB ∪ BA = G]. The study of this event gives rise to a group invariant we call Θ(G). Θ(G) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes √Θ(G)|G| log |G|; i.e. for any ɛ > 0, if the size of A and B is less than (1 − ɛ)√Θ(G)|G| log |G|, then with high probability AB ∪ BA ≠ G. If A and B are larger than (1 + ɛ)√Θ(G)|G| log |G|, then AB ∪ BA = G with high probability.     

Additive Latin transversals and group rings

LetA={a ₁, …,a _k} and {b ₁, …,b _k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata _i+b _i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to the groupsZ _p ^r andZ _p ^rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|. This work has been supported partly by NSFC grant number 19971058 and 10271080.     

NON-EMPTY CROSS-2-INTERSECTING FAMILIES OF SUBSETS

Let Cdenote the set of all k-subests of an n-set.Assume Alohtain in Ca,and A lohtain in (A,B) is called a cross-2-intersecting family if |A B≥2 for and A∈A,B∈B.In this paper,the best upper bounds of the cardinalities for non-empty cross-2-intersecting familles of a-and b-subsets are obtained for some a and b,A new proof for a Frankl-Tokushige theorem[6] is also given.     

Bounds on global secure sets in cactus trees

Let G = (V, E) be a graph. A global secure set SD ⊆ V is a dominating set which satisfies the condition: for all X ⊆ SD, |N[X] ∩ SD| ≥ | N[X] − SD|. A global defensive alliance is a set of vertices A that is dominating and satisfies a weakened condition: for all x ∈ A, |N[x] ∩ A| ≥ |N[x] − A|. We give an upper bound on the cardinality of minimum global secure sets in cactus trees. We also present some results for trees, and we relate them to the known bounds on the minimum cardinality of global defensive alliances.     

Transversals of additive Latin squares

LetA={a ₁, …,a _k} andB={b ₁, …,b _k} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈S _ksuch that the sums α _i +b _i , 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method in various finite and infinite fields we extend Alon’s result to the groups (ℤ _p ) ^a and in the casek<p, and verify Snevily’s conjecture for every cyclic group of odd order. Supported by Hungarian research grants OTKA F030822 and T029759. Supported by the Catalan Research Council under grant 1998SGR00119. Partially supported by the Hungarian Research Foundation (OTKA), grant no. T029132.     

A note on Freiman models in Heisenberg groups

Green and Ruzsa recently proved that for any s ≥ 2, any small squaring set A in a (multiplicative) abelian group, i.e., |A·A| < K|A|, has a Freiman smodel: it means that there exists a group G and a Freiman s-isomorphism from A into G such that |G| < f (s,K)|A|.     

A measuring argument for finite permutation groups

LetG be a finite transitive permutation group on a finite setS. LetA be a nonempty subset ofS and denote the pointwise stabilizer ofA inG byC _G (A). Our main result is the following inequality: [G :C _G (A)]≥|G|^|A|/|S|. This paper is a part of the author’s Ph.D. thesis research, carried out at Tel Aviv University under the supervision of Professor Marcel Herzog.     

Factorization for non-nevanlinna classes of analytic functions

A generalization of the Blaschke product is constructed. This product enables one to factor out the zeros of the members of certain non-Nevanlinna classes of functions analytic in the unit disc, so that the remaining (non-vanishing) functions still belong to the same class. This is done for the classesA ⁻ⁿ (0<n<∞) andB ⁻ⁿ (0<n<2) defined as follows:f ∈A ⁻ⁿ iff |f(z)|≦C _f (1−|z|)⁻ⁿ ,f ∈B ⁻ⁿ iff |f(z)|≦exp {C _f (1−|z|)⁻ⁿ }, whereC _f depends onf.     

The rectifiability threshold in abelian groups

For any abelian group G and integer t ≥ 2 we determine precisely the smallest possible size of a non-t-rectifiable subset of G. Specifically, assuming that G is not torsion-free, denote by p the smallest order of a non-zero element of G. We show that if a subset S ⊆ G satisfies |S| ≤ ⌌log _t p⌍, then S is t-isomorphic (in the sense of Freiman) to a set of integers; on the other hand, we present an example of a subset S ⊆ G with |S| = ⌌log _t p⌍ + 1 which is not t-isomorphic to a set of integers.     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

Algebras whose equivalence relations are congruences

It is proved that all the equivalence relations of a universal algebra A are its congruences if and only if either |A| ≤ 2 or every operation f of the signature is a constant (i.e., f(a ₁ , . . . , a _n ) = c for some c ∈ A and all the a ₁ , . . . , a _n ∈ A) or a projection (i.e., f(a ₁ , . . . , a _n ) = a _i for some i and all the a ₁ , . . . , a _n ∈ A). All the equivalence relations of a groupoid G are its right congruences if and only if either |G| ≤ 2 or every element a ∈ G is a right unit or a generalized right zero (i.e., x _a  = y _a for all x, y ∈ G). All the equivalence relations of a semigroup S are right congruences if and only if either |S| ≤ 2 or S can be represented as S = A∪B, where A is an inflation of a right zero semigroup, and B is the empty set or a left zero semigroup, and ab = a, ba = a ² for a ∈ A, b ∈ B. If G is a groupoid of 4 or more elements and all the equivalence relations of it are right or left congruences, then either all the equivalence relations of the groupoid G are left congruences, or all of them are right congruences. A similar assertion for semigroups is valid without the restriction on the number of elements.     

Table algebras with multiple P-polynomial structures

Using covering numbers we prove that a standard real integral table algebra (A, B) with |B| ≥ 6 has a P-polynomial structure with respect to every b ≠ 1 in B if and only if 2|B|-1 is prime and (A, B) is exactly isomorphic to the Bose-Mesner algebra of the association scheme of the ordinary (2|B|-1)-gon. Then we present an example showing that this result is not true if |B| ≤ 5.     

A Neighborhood Condition for Graphs to Have [a, b]-Factors II

 Let a, b, m, and t be integers such that 1≤a<b and 1≤t≤⌉(b−m+1)/a⌉. Suppose that G is a graph of order |G| and H is any subgraph of G with the size |E(H)|=m. Then we prove that G has an [a,b]-factor containing all the edges of H if the minimum degree is at least a, |G|>((a+b)(t(a+b−1)−1)+2m)/b, and |N _G (x ₁)∪⋯ ∪N _G (x _t )|≥(a|G|+2m)/(a+b) for every independent set {x ₁,…,x _t }⊆V(G). This result is best possible in some sense and it is an extension of the result of H. Matsuda (A neighborhood condition for graphs to have [a,b]-factors, Discrete Mathematics 224 (2000) 289–292). Received: October, 2001 Final version received: September 17, 2002 RID="*" ID="*" This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 13740084, 2001     

The riesz turndown collar theorem giving an asymptotic estimate of the powers of an operator under the ritt condition

For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)⁻¹‖≤C|z−1|⁻¹, |z|>1, we show how to use the Riesz turndown collar theorem to estimate sup _n≥0‖T ⁿ‖. A similar estimate is shown for lim sup _n ‖T ⁿ‖ in terms of the Ritt constantM=lim sup _z→1‖(1−z)(zI−T)⁻¹‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤C _q‖f‖ _Mult , where ‖·‖ _Mult stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1. Notation.D denotes the open unit disc of the complex plane,D={z∈ℂ:|z|<1}, andT={z∈ℂ:|z|=1} is the unit circle.H ^∞ is the Banach algebra of bounded analytic functions onD equipped with the supremum norm ‖.‖_∞.