Torsion units in integral group rings of metacyclic groups

It is proved that if G is a split extension of a cyclic p-group by a cyclic p′-group with faithful action then any torsion unit of augmentation one of $\text{Z}$G is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (|A|, |X|) = 1 then any torsion unit of $\text{Z}$G of augmentation one and order relatively prime to |A| is rationally conjugate to an element of X.     

Non-commutative operator Bohr inequality

It is shown that if A, B, X are Hilbert space operators such that X?γI, for the positive real number γ, and p,q>1 with 1/p+1/q=1, then |A−B|²?p|A|²+q|B|² with equality if and only if (1−p)A=B and γ||||A−B|²|||?|||p|A|²X+qX|B|²||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert-Schmidt operator, then ‖δ_A,B²(X)‖₂?‖p|A|²X+qX|B|²‖₂ with equality if and only if (1−p)AX=XB.     

Relative ranks of Lipschitz mappings on countable discrete metric spaces

Let X be a countable discrete metric space and let XX denote the family of all functions on X. In this article, we consider the problem of finding the least cardinality of a subset A of XX such that every element of XX is a finite composition of elements of A and Lipschitz functions on X. It follows from a classical theorem of Sierpiński that such an A either has size at most 2 or is uncountable.We show that if X contains a Cauchy sequence or a sufficiently separated, in some sense, subspace, then |A|≤1. On the other hand, we give several results relating |A| to the cardinal d; defined as the minimum cardinality of a dominating family for NN. In particular, we give a condition on the metric of X under which |A|≥d holds and a further condition that implies |A|≤d. Examples satisfying both of these conditions include all subsets of Nk and the sequence of partial sums of the harmonic series with the usual euclidean metric.To conclude, we show that if X is any countable discrete subset of the real numbers R with the usual euclidean metric, then |A|=1 or almost always, in the sense of Baire category, |A|=d.     

Invariant subspaces in Bergman spaces and Hedenmalm's boundary value problem

A functionG in a Bergman spaceA ^p , 0<p<∞, in the unit diskD is calledA ^p -inner if |G| ^p ?1 annihilates all bounded harmonic functions inD. Extending a recent result by Hedenmalm forp=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $$\Delta \Phi = \chi _D (|G|^p - 1),$$ where χ _D denotes the characteristic function ofD andG is an arbitraryA ^p -inner function, is continuous inC, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman spaceA ² to a generalA ^p -setting.     

Component factors and induced subgraphs

Let G and K be connected graphs such that | G| = n|K| (n ≥ 2) and let p be a fixed integer satisfying 1 < p < n. We prove that if G \ A has a K-factor for every connected subgraph A with |A| = p|K|, then G also has a K-factor. © 1996 John Wiley & Sons, Inc.     

Strongly dependent theories

We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ? ? and |I|≥?_|T|+(µ), then some J ? I of cardinality µ⁺ is an indiscernible sequence over A.     

On finite groups with relatively large centralizers of invariant subgroups

In the paper, the finite groups G are studied for which every invariant subgroup A has the property that |G: AC _G (A)| divides a fixed prime p.     

Eigenvalues of Finite Projective Planes with an Abelian Cartesian Group

Let M be an incidence matrix for a projective plane of order n. The eigenvalues of M are calculated in the Desarguesian case and a standard form for M is obtained under the hypothesis that the plane admits a (P,L)-transitivity G, |G| = n. The study of M is reduced to a principal submatrix A which is an incidence matrix for n ² lines of an associated affine plane. In this case, A is a generalized Hadamard matrix of order n for the Cayley permutation representation R(G). Under these conditions it is shown that G is a 2-group and n = 2^r when the eigenvalues of A are real. If G is abelian, the characteristic polynomial |xI – A| is the product of the n polynomials |x – (A)|, a linear character of G. This formula is used to prove n is a prime power under natural conditions on A and spectrum(A). It is conjectured that |xI – A| x ⁿ² mod p for each prime divisor p of n and the truth of the conjecture is shown to imply n = |G| is a prime power.     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.     

Probability that the commutator of two group elements is equal to a given element

In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which |G^′| is prime and G^′≤Z(G) as well as for groups G which |G^′| is prime and G^′∩Z(G)=1. This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237-247].     

Elementary abelian operator groups

Letp be a prime,n a positive integer. Suppose thatG is a finite solvablep'-group acted on by an elementary abelianp-groupA. We prove that ifC _G (ϕ) is of nilpotent length at mostn for every nontrivial element ϕ ofA and |A|≥p ⁿ⁺¹ thenG is of nilpotent length at mostn+1.     

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.     

Note: A conjecture on partitions of groups

We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A _n A _n ^?1 ) = |G| for each n ∈ ω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

Perfect Matchings in Balanced Hypergraphs – A Combinatorial Approach

Our topic is an extension of the following classical result of Hall to hypergraphs: A bipartite graph G contains a perfect matching if and only if for each independent set X of vertices, at least |X| vertices of G are adjacent to some vertex of X. Berge generalized the concept of bipartite graphs to hypergraphs by defining a hypergraph G to be balanced if each odd cycle in G has an edge containing at least three vertices of the cycle. Based on this concept, Conforti, Cornuéjols, Kapoor, and Vušković extended Hall's result by proving that a balanced hypergraph G contains a perfect matching if and only if for any disjoint sets A and B of vertices with |A| > |B|, there is an edge in G containing more vertices in A than in B (for graphs, the latter condition is equivalent to the latter one in Hall's result). Their proof is non-combinatorial and highly based on the theory of linear programming. In the present paper, we give an elementary combinatorial proof. Received April 29, 1997     

Classifying Cubic Semisymmetric Graphs of Order 18 p n

A graph X is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut (X) acting transitively on its edge set but not on its vertex set. In the case of G = A, we call X a semisymmetric graph. Let p be a prime. It was shown by Folkman (J Comb Theory 3:215–232, 1967) that a regular edge-transitive graph of order 2p or 2p ² is necessarily vertex-transitive. The smallest semisymmetric graph is the Folkman graph. In this study, we classify all connected cubic semisymmetric graphs of order 18p ⁿ , where p is a prime and \({n \geq 1}\) .     

On sets with small doubling property

Suppose that G is an arbitrary Abelian group and A is any finite subset G. A set A is called a set with small sumset if, for some number K, we have |A + A| ≤ K|A|. The structural properties of such sets were studied in the papers of Freiman, Bilu, Ruzsa, Chang, Green, and Tao. In the present paper, we prove that, under certain constraints on K, for any set with small sumset, there exists a set Λ, Λ ? _? K log |A|, such that |A ν Λ| ? |A|/K ^1/2+? , where ? > 0. In contrast to the results of the previous authors, our theorem is nontrivial even for a sufficiently large K. For example, for K we can take |A| ^η , where η > 0. The method of proof used by us is quite elementary.     

Finite <Emphasis Type="Italic">p</Emphasis>-groups all of whose maximal abelian subgroups are soft

A subgroup A of a p-group G is said to be soft in G if C _G (A) = A and |N _G (A/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.     

Minimum degree of bipartite graphs and the existence ofk-factors

LetG be a bipartite graph with bipartition (X, Y) andk a positive integer. If (i) $$\left| X \right| = \left| Y \right|,$$ (ii) $$\delta (G) \geqslant \left\lceil {\frac{{\left| X \right|}}{2}} \right\rceil \geqslant k,$$ \(\left| X \right| \geqslant 4k - 4\sqrt k + 1\) when |X| is odd and |X| ≥ 4k ? 2 when |X| is even, thenG has ak-factor.