On galois groups and their maximal 2-subgroups

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL ₁ of ℚ(T) with Galois groupG. LetL be the subfield ofL ₁ fixed byH. We make the hypothesis thatL ₁ admits a quadratic extensionL ₂ which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL ₁ then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL ₂. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL ₁ admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).     

Proof of a conjecture of T. Gallai concerning connectivity properties of colour-critical graphs

Tibor Gallai made the following conjecture. LetG be ak-chromatic colour-critical graph. LetL denote the set of those vertices ofG having valencyk−1 and letH be the rest ofV(G). Then the number of components induced byL is not less than the number of components induced byH, providedL ≠ 0. We prove this conjecture in a slightly generalized form. Dedicated to Tibor Gallai on his seventieth birthday     

An unfeasible matching problem

LetG be a bipartite graph with natural edge weights, and letW be a function from the set of vertices ofG into natural numbers. AW-matching ofG is a subset of the set of edges ofG such that for each vertexv the total weight of edges in the subset incident tov does not exceedW(v). Letm be a natural number. We show that the problem of deciding whether there is aW-matching inG whose total weight is not less thanm is NP-complete even ifG is bipartite and its edge weights as well as theW(v)-constraints are constantly bounded.     

On equivalence of number fields

LetK be a field,G a finite group.G is calledK-admissible iff there exists a finite dimensionalK-central division algebraD which is a crossed product forG. Now letK andL be two finite extensions of the rationalsQ such that for every finite groupG, G isK-admissible if and only ifG isL-admissible. ThenK andL have the same degree and the same normal closure overQ. An erratum to this article is available at .     

Perturbation bounds for the Cholesky andQR factorizations

LetA, A+E be Hermitian positive definite matrices. Suppose thatA=LL ^H andA+E=(L+G)(L+G)^H are the Cholesky factorizations ofA andA+E, respectively. In this paper lower bounds and upper bounds on |G|/|L| in terms of |E|/|A| are given. Moreover, perturbation bounds are given for the QR factorization of a complexm ×n matrixA of rankn.This research was supported by the National Science Foundation of China and the Department of Mathematics of Linköping University in Sweden.     

Frattini closed groups and adequate extensions of global fields

LetL be a finite Galois extension of a global fieldF. It is shown that if the Galois groupG=Gal(L/F) satisfies a certain condition, thenL is a maximal commutative subfield of someF-division algebra if and only if the intermediate field corresponding to the Frattini subgroup ofG is also a maximal commutative subfield of someF-division algebra. In particular this condition holds ifG is a supersolvable group. The third author was supported in part by the NSF under Grant DMS 97-01253.     

On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L) ^G ]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A ^x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.     

Multipliers and Modulus on Banach Algebras Related to Locally Compact Groups

LetGbe a locally compact group. In this paper we study moduli of products of elements and of multipliers of Banach algebras which are related to locally compact groups and which admit lattice structure. As a consequence, we obtain a characterization of operators onL^∞(G) which commute with convolutions whenGis amenable as discrete.     

Families of finite sets with three intersections

LetX be a finite set ofn elements and ℓ a family ofk-subsets ofX. Suppose that for a given setL of non-negative integers all the pairwise intersections of members of ℓ have cardinality belonging toL. Letm(n, k, L) denote the maximum possible cardinality of ℓ. This function was investigated by many authors, but to determine its exact value or even its correct order of magnitude appears to be hopeless. In this paper we investigate the case |L|=3. We give necessary and sufficient conditions form(n, k, L)=O(n) andm(n, k, L)≧O(n ²), and show that in some casesm(n, k, L)=O(n ^3/2), which is quite surprising.     

Automorphisms of finite groups of bounded rank

LetG be a finite group admitting an automorphismα withm fixed points. Suppose every subgroup ofG isr-generated. It is shown that (1)G has a characteristic soluble subgroupH whose index is bounded in terms ofm andr, and (2) if the orders ofα andG are coprime, then the derived length ofH is also bounded in terms ofm andr. To Professor John Thompson, in honor of his outstanding achievements     

On automorphism groups of Cayley graphs

LetX _G,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(X_G,H) must contain the regular subgroupL _G corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(X_G,H)L_G] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.     

A reciprocity theorem for unitary representations of Lie groups

LetG be a Lie group,H a closed subgroup,L a unitary representation ofH andU ^L the corresponding induced representation onG. The main result of this paper, extending Ol’ŝanskii’s version of the Frobenius reciprocity theorem, expresses the intertwining number ofU ^L and an irreducible unitary representationV ofG in terms ofL and the restriction ofV _∞ toH.     

TheK-admissibility of 2A 6 and 2A 7

LetK be a field and letG be a finite group.G isK-admissible if there exists a Galois extensionL ofK withG=Gal(L/K) such thatL is a maximal subfield of a centralK-division algebra. This paper contains a characterization of those number fields which areQ ₁₆-admissible. This is the same class of number fields which are 2A ₆=SL(2, 9) and 2A ₇ admissible. Dedicated to John Thompson to celebrate his Wolf Prize in Mathematics 1992     

Maximality of compression semigroups

LetG be a connected semisimple Lie group andr an involution onG. Further letL be an open subgroup of the groupG ^r ofr-fixed points andP⊂-G a parabolic subgroup. The semigroupS(L,P)∶={g∈G∶gLP⊂-LP} is called the compression semigroup of theL-orbit of the base point in the flag manifoldG/P. We show that compression semigroups for open orbits and regular symmetric pairs are maximal semigroups. Supported by a DFG Heisenberg-grant.     

Convolution quotients of nonnegative functions

LetG be a locally compact commutative Hausdorff group andf a function belonging toL ₁(G). If the integral off with respect to the Haar measure is positive, then one can find a nonnegative (not identically 0) functiong such that the convolution off andg is also nonnegative.     

On primary ideals in group algebras

LetG be a -compact locally compact group such thatG/Z(G), whereZ(G) denotes the center ofG, has a relatively compact commutator subgroup. It is shown that primary ideals inL ¹(G) are maximal.     

On certain homomorphisms of restriction algebras of symmetric sets

LetG be a locally compact abelian group and Γ its dual group. For any closedH ⊆G denote the algebra of restrictions toH of Fourier transforms of functions inL ¹(Γ) byA(H). This paper considers certain Cantor like sets inR and ΠZ _m(j) and gives some necessary algebraic criterion fornatural isomorphisms of their restriction algebras. This work was supported mainly by the U.S. National Science Foundation Graduate Fellowship Program. The author wishes to thank Paul Cohen, Karel de Leeuw, and Yitzhak Katznelson for their counsel.     

Construction of markov process with markov resolvent onL P (E, m)

LetE be a Hausdorff topological space and let β(E) be its Borel σ-field. Letm be a σ-finite measure on (E, β(E)). A necessary and sufficient condition for a Markov resolvent onL ^P (E,m) to be associated with anm-tightm-special standard process (with state spaceE) is given. Furthermore some new examples which do not belong to the framework of Dirichlet space are also given. Project supported by the Youth Foundation of Chinese Academy of Sciences.     

(g,f)-Factorizations of graphs orthogonal to [1,2]-subgraph

LetG be a simple graph. Letg(x) andf(x) be integer-valued functions defined onV(G) withf(x)g(x)1 for allxV(G). It is proved that ifG is an (mg+m–1,mf–m+1)-graph andH is a [1,2]-subgraph withm edges, then there exists a (g,f)-factorization ofG orthogonal toH.This work is supported by China Postdoctoral Science Foundation and Shandong Youth Science Foundation.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.