A characterization theorem for certain unions of two starshaped sets in R 2

Let S be a compact set in R ². For S simply connected, S is a union of two starshaped sets if and only if for every F finite, F bdry S, there exist a set G bdry S arbitrarily close to F and points s, t depending on G such that each point of G is clearly visible via S from one of s, t. In the case where S has at most finitely many components, the necessity of the condition still holds while the sufficiency fails.     

Covering and packing in \Bbb Zn{\Bbb Z}^n and \Bbb Rn{\Bbb R}^n, (I)

Given a finite subset A{\cal A} of an additive group \Bbb G{\Bbb G} such as \Bbb Zⁿ{\Bbb Z}^n or \Bbb Rⁿ{\Bbb R}^n , we are interested in efficient covering of \Bbb G{\Bbb G} by translates of A{\cal A} , and efficient packing of translates of A{\cal A} in \Bbb G{\Bbb G} . A set S ì \Bbb G{\cal S} \subset {\Bbb G} provides a covering if the translates A + s{\cal A} + s with s ? Ss \in {\cal S} cover \Bbb G{\Bbb G} (i.e., their union is \Bbb G{\Bbb G} ), and the covering will be efficient if S{\cal S} has small density in \Bbb G{\Bbb G} . On the other hand, a set S ì \Bbb G{\cal S} \subset {\Bbb G} will provide a packing if the translated sets A + s{\cal A} + s with s ? Ss \in {\cal S} are mutually disjoint, and the packing is efficient if S{\cal S} has large density. In the present part (I) we will derive some facts on these concepts when \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and give estimates for the minimal covering densities and maximal packing densities of finite sets A ì \Bbb Zⁿ{\cal A} \subset {\Bbb Z}^n . In part (II) we will again deal with \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and study the behaviour of such densities under linear transformations. In part (III) we will turn to \Bbb G = \Bbb Rⁿ{\Bbb G} = {\Bbb R}^n .     

Factorization in the extended Selberg class of L ‐functions associated with holomorphic modular forms

Let N ∈ ? and let χ be a Dirichlet character modulo N. Let f be a modular form with respect to the group Γ₀(N), multiplier χ and weight k. Let F be the L ‐function associated with f and normalized in such a way that F (s) satisfies a functional equation where s reflects in 1 – s. The modular forms f for which F belongs to the extended Selberg class S^# are characterized. For these forms the factorization of F in primitive elements of S^# is enquired. In particular, it is proved that if f is a cusp form and F ∈ S^# then F is almost primitive (i.e., that if F = PG is a factorization with P, G ∈ S^# and the degree of P is < 2 then P is a Dirichlet polynomial). It is also proved that the conductor of the polynomial factor P is bounded by N. If f belongs to the space generated by newforms and N ≤ 4 then F is actually primitive (i.e., P is a constant) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Group Actions,k-Derivations,and Finite Morphisms

Let G be an affine algebraic group over an algebraically closed field k of characteristic zero. In this article, we consider finite G-equivariant morphisms F:X → Y of irreducible affine G-varieties. First we determine under which conditions on Y the induced map F ^G :X//G → Y//G of quotient varieties is also finite. This result is reformulated in terms of kernels of derivations on k-algebras A ? B such that B is integral over A. Second we construct explicitly two examples of finite G-equivariant maps F. In the first one, F ^G is quasifinite but not finite. In the second one, F ^G is not even quasifinite.     

Generation of exceptional groups of Lie-type

In this paper we prove two theorems concerning the generation of a finite exceptional group of Lie-type G _F. The first is: there is a semisimple element s such that for nearly all elements x G _Fthe elements s and x generate the group G _F. The second theorem we prove is: if G is a finite simple exceptional group of Lie-type not of type E ₆ or ² E ₆, then it is generated by three involutions.The author gratefully acknowledges financial support by the Deutsche Forschungs-gemeinschaft.     

On some medial semigroups with an associate subgroup

Let S be a semigroup and s,t∈S. We say that t is an associate of s if s=sts. If S has a maximal subgroup G such that every element s of S has a unique associate in G, say s ^∗, we say that G is an associate subgroup of S and consider the mapping s→s ^∗ as a unary operation on S. In this way, semigroups with an associate subgroup may be identified with unary semigroups satisfying three simple axioms. Among them, only those satisfying the identity (st)^∗=t ^∗ s ^∗, called medial, have a structure theorem, due to Blyth and Martins.     

Cubic vertex‐transitive graphs of order 2pq

A graph is vertex?transitive or symmetric if its automorphism group acts transitively on vertices or ordered adjacent pairs of vertices of the graph, respectively. Let G be a finite group and S a subset of G such that 1?S and S={s^?1 | s∈S}. The Cayleygraph Cay(G, S) on G with respect to S is defined as the graph with vertex set G and edge set {{g, sg} | g∈G, s∈S}. Feng and Kwak [J Combin Theory B 97 (2007), 627–646; J Austral Math Soc 81 (2006), 153–164] classified all cubic symmetric graphs of order 4p or 2p² and in this article we classify all cubic symmetric graphs of order 2pq, where p and q are distinct odd primes. Furthermore, a classification of all cubic vertex‐transitive non‐Cayley graphs of order 2pq, which were investigated extensively in the literature, is given. As a result, among others, a classification of cubic vertex‐transitive graphs of order 2pq can be deduced. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 285–302, 2010     

On graph equivalences preserved under extensions

Let G be the set of finite graphs whose vertices belong to some fixed countable set, and let ≡ be an equivalence relation on G. By the strengthening of ≡ we mean an equivalence relation ≡_s such that G≡_sH, where G,H∈G, if for every F∈G, G∪F≡H∪F. The most important case that we study in this paper concerns equivalence relations defined by graph properties. We write G≡^ΦH, where Φ is a graph property and G,H∈G, if either both G and H have the property Φ, or both do not have it. We characterize the strengthening of the relations ≡^Φ for several graph properties Φ. For example, if Φ is the property of being a k-connected graph, we find a polynomially verifiable (for k fixed) condition that characterizes the pairs of graphs equivalent with respect to . We obtain similar results when Φ is the property of being k-colorable, edge 2-colorable, Hamiltonian, or planar, and when Φ is the property of containing a subgraph isomorphic to a fixed graph H. We also prove several general theorems that provide conditions for ≡_s to be of some specific form. For example, we find a necessary and sufficient condition for the relation ≡_s to be the identity. Finally, we make a few observations on the strengthening in a more general case when G is the set of finite subsets of some countable set.     

Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov

Let G be a locally finite group satisfying the condition given in the title and suppose that G is not nilpotent-by-Chernikov. It is shown that G has a section S that is not nilpotent-by-Chernikov, where S is either a p-group or a semi-direct product of the additive group A of a locally finite field F by a subgroup K of the multiplicative group of F, where K acts by multiplication on A and generates F as a ring. Non-(nilpotent-by-Chernikov) extensions of this latter kind exist and are described in detail.     

A Classification of Fully Residually Free Groups of Rank Three or Less

A groupGisfully residually freeprovided to every finite setS ⊂ G\{1} of non-trivial elements ofGthere is a free groupF_Sand an epimorphismh_S:G → F_Ssuch thath_S(g) ≠ 1 for allg ∈ S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free.     

The number of labeled graphs placeable by a given permutation

Let S be a finite set and σ a permutation on S. The permutation σ* on the set of 2-subsets of S is naturally induced by σ. Suppose G is a graph and V(G), E(G) are the vertex set, the edge set, respectively. Let V(G) = S. If E(G) and σ*(E(G)), the image of E(G) by σ*, have no common element, then G is said to be placeable by σ. This notion is generalized as follows. If any two sets of {E(G), (σ¹)*(E(G)),…,(σ^l−1)* (E(G))} have no common element, then G is said to be I-placeable by σ. In this paper, we count the number of labeled graphs which are I-placeable by a given permutation. At first, we introduce the interspaced Ith Fibonacci and Lucas numbers. When I = 2 these numbers are the ordinary Fibonacci and Lucas numbers. It is known that the Fibonacci and Lucas numbers are rounded powers. We show that the interspaced Ith Fibonacci and Lucas numbers are also rounded powers when I = 3. Next, we show the number of labeled graphs which are I-placeable by a given permutation is a product of the interspaced Ith Lucas numbers. Finally, using a property of the generalized binomial series, we count the number of labeled graphs of size k which are I-placeable by σ. © 1996 John Wiley & Sons, Inc.     

GB-Problem in the Class of Locally Graded Groups

A problem, we consider, is equivalent to the one posed in 1981 by Bergman: Let G be a group and S a subsemigroup of G which generates G as a group. Must each identity satisfied in S be satisfied in G? The first counterexample was found in 2005 by Ivanov and Storozhev. It gives a negative answer to the problem in general. However we show that the problem has an affirmative answer for locally residually finite groups and for locally graded groups containing no free noncyclic subsemigroups.     

Automorphic Plancherel density theorem

Let F be a totally real field, G a connected reductive group over F, and S a finite set of finite places of F. Assume that G(F ?_? ?) has a discrete series representation. Building upon work of Sauvageot, Serre, Conrey-Duke-Farmer and others, we prove that the S-components of cuspidal automorphic representations of $G\left( {\mathbb{A}_F } \right)$ are equidistributed with respect to the Plancherel measure on the unitary dual of G(F _S ) in an appropriate sense. A few applications are given, such as the limit multiplicity formula for local representations in the global cuspidal spectrum and a quite flexible existence theorem for cuspidal automorphic representations with prescribed local properties. When F is not a totally real field or G(F ?_? ?) has no discrete series, we present a weaker version of the above results.     

Commutative Schur Rings of Maximal Dimension

A commutative Schur ring over a finite group G has dimension at most s _G  = d ₁ + … +d _r , where the d _i are the degrees of the irreducible characters of G. We find families of groups that have S-rings that realize this bound, including the groups SL(2, 2 ⁿ ), metacyclic groups, extraspecial groups, and groups all of whose character degrees are 1 or a fixed prime. We also give families of groups that do not realize this bound. We show that the class of groups that have S-rings that realize this bound is invariant under taking quotients. We also show how such S-rings determine a random walk on the group and how the generating function for such a random walk can be calculated using the group determinant.     

The McKay conjecture for exceptional groups and odd primes

Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G → G and G := G ^F , such that the root system is of exceptional type or G is a Suzuki group or Steinberg’s triality group. We show that all irreducible characters of C _G (S), the centraliser of S in G, extend to their inertia group in N _G (S), where S is any F-stable Sylow torus of (G, F). Together with the work in [16] this implies that the McKay conjecture is true for G and odd primes ℓ different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [14]. This research has been supported by the DFG-grant “Die Alperin-McKay-Vermutung für endliche Gruppen” and an Oberwolfach Leibniz fellowship.     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

Constrained Ramsey numbers of graphs

Given two graphs G and H, let f(G,H) denote the minimum integer n such that in every coloring of the edges of K_n, there is either a copy of G with all edges having the same color or a copy of H with all edges having different colors. We show that f(G,H) is finite iff G is a star or H is acyclic. If S and T are trees with s and t edges, respectively, we show that 1+s(t?2)/2≤f(S,T)≤(s?1)(t²+3t). Using constructions from design theory, we establish the exact values, lying near (s?1)(t?1), for f(S,T) when S and T are certain paths or star‐like trees. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 1–16, 2003     

Actions of Inverse Semigroups on Algebras

In this article we prove that if S is a faithfully projective R-algebra and H is a finite inverse semigroup acting on S as R-linear maps such that the fixed subring S ^H  = R, then any partial isomorphism between ideals of S which are generated by central idempotents can be obtained as restriction of an R-automorphism of S and there exists a finite subgroup of automorphisms G of S with S ^G  = R.     

Maximal Crossed Product Orders Over Discrete Valuation Rings

The problem of determining when a (classical) crossed product T = S ^f ?G of a finite group G over a discrete valuation ring S is a maximal order, was answered in the 1960s for the case where S is tamely ramified over the subring of invariants S ^G . The answer was given in terms of the conductor subgroup (with respect to f) of the inertia. In this article we solve this problem in general when S/S ^G is residually separable. We show that the maximal order property entails a restrictive structure on the subcrossed product graded by the inertia subgroup. In particular, the inertia is abelian. Using this structure, one is able to extend the notion of the conductor. As in the tame case, the order of the conductor is equal to the number of maximal two-sided ideals of T and hence to the number of maximal orders containing T in its quotient ring. Consequently, T is a maximal order if and only if the conductor subgroup is trivial.     

Nilpotent orbits over ground fields of good characteristic

Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in (F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge. Under some further (fairly mild) assumptions on G, we prove convergence for arbitrary orbital integrals on the Lie algebra and on the group. The convergence of orbital integrals in the case where F has characteristic 0 was obtained by Deligne and Ranga Rao (1972).