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.     

Covers of varieties with completely split points

LetK be a field such that all Sylow subgroups of its absolute Galois groupG _Kare infinite. LetX be a smooth variety overK with function fieldF andY→X the normalisation in a finite, separable extensionE/vbF. We show: If there is a closed pointx∈X which does not split completely inY→X, then the set of these points is Zariski dense inX.     

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).     

On theG-matrices with entries and eigenvalues inQ(i)

LetG be a finite abelian group,K a subfield ofC, C[G] regarded as an algebra of matrices.A _G ^K {AC[G]| all the entries and eigenvalues ofA are inK} is an association algebra overK. In this paper, the association scheme ofA _G ^K is determined and in the caseK=Q(i), the first eigenmatrix of the association scheme computed. As an application, it is proved thatZ ₄×Z ₄×Z ₄ is the only abelian group admitted as a Singer group by some distance-regular digraph of girth 4 on 64 vertices.     

Galois extensions with bounded ramification in characteristicp on a question of S. Abhyankar

Given an arbitrary congruence function fieldK of characteristicp and a finite groupG with a uniquep-Sylow subgroupp(G) which is abelian and for which the factor groupG/p(G) is nilpotent and hass generators, there exists a geometric Galois extensionL/K with Galois groupG in which preciselys prime divisors ofK are ramified.     

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 .     

On the galois module structure of semisimple holomorphic differentials

LetL/K be a finite Galoisp-extension of algebraic function fields of one variable over an algebraically closed fieldk of characteristicp, with Galois groupG=Gal(L/K). The space Ώ _L ^s (0) of semisimple holomorphic differentials ofL is thek-vector space of holomorphic differentials which are fixed by the Cartier operator. We obtain the isomorphism classes and multiplicities of the summands in a Krull-Schmidt decomposition of thek[G]-module Ώ _L ^s (0) into a direct sum of indecomposablek[G]-modules. Partially supported by CONACyT, project No. 25063-E.     

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 finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

Q-admissibility of certain nonsolvable groups

A finite groupG is calledQ-admissible if there exists a finite dimensional central division algebra overQ, containing a maximal subfield which is a Galois extension ofQ with Galois group isomorphic toG. It is proved thatS ₅ ⁻ , one of the two nontrivial central extensions ofS ₅ byZ/2Z, isQ-admissible. As a consequence of that result and previous results of Sonn and Stern, every finite Sylow-metacyclic group, havingA ₅ as a composition factor, isQ-admissible. This paper is part of a M.Sc. thesis written at the Technion — Israel Institute of Technology, under the supervision of Professor J. Sonn, whom the author wishes to thank for his valuable guidance.     

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.     

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     

On free quadratic extensions of rings

The quaternion algebraB[j] over a commutative ringB with 1 defined byS. Parimala andR. Sridharan is generalized in two directions: (1) the ringB may be non-commutative with 1, and (2)j ² may be any invertible element (not necessarily –1). LetG={} be an automorphism group ofB of order 2, andA={b inB| (b)=b}. LetB[j] be a generalized quaternion algebra such thataj (a) for eacha inB. It will be shown thatB is Galois (for non-commutative ring extensions) overA which is contained in the center ofB if and only ifB[j] is Azumaya overA. Also,A[j] is a splitting ring forB[j] such thatA[j] is Galois overA. Moreover, we shall determine which automorphism group ofA[j] is a Galois group.     

Über Bewertungen,welche unter einer reduktiven Gruppe invariant sind

Summary LetG be a reductive group defined over an algebraically closed fieldk and letX be aG-variety. In this paper we studyG-invariant valuationsv of the fieldK of rational functions onX. These objects are fundamental for the theory of equivariant completions ofX. LetB be a Borel subgroup andU the unipotent radical ofB. It is proved thatv is uniquely determined by its restriction toK ^U . Then we study the set of invariant valuations having some fixed restrictionv ₀, toK ^B . Ifv ₀ is geometric (i.e., induced by a prime divisor) then this set is a polyhedron in some vector space. In characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupW ^X . Thus, the classification of invariant valuations is almost reduced to the classification of valuations ofK ^B .

Unterstützt durch den Schweizerischen Nationalfonds zur Förderung der wissenschaftlichen Forschung.     

Universal finite group extensions and a non-splitting theorem

LetG andK be finite groups whose orders have a common prime divisor. Then there is a groupK ^* closely related toK for which there is a non-split extension ofK ^* byG. I wish to express thanks to the Mathematics Institute of the Hebrew University of Jerusalem for its hospitality from September to December 1972, and to Dr. Avinoam Mann for his helpful comments.     

Lower degree bounds for modular invariants and a question of I. Hughes

We prove two statements. The first one is a conjecture of Ian Hughes which states that iff ₁, ..., f_n are primary invariants of a finite linear groupG, then the least common multiple of the degrees of thef _i is a multiple of the exponent ofG.The second statement is about vector invariants: IfG is a permutation group andK a field of positive characteristicp such thatp divides |G|, then the invariant ringK[V ^m]^G ofm copies of the permutation moduleV overK requires a generator of degreem(p–1). This improves a bound given by Richman [6], and implies that there exists no degree bound for the invariants ofG that is independent of the representation.     

Quantum homogeneous spaces with faithfully flat module structures

LetA be a Hopf algebra with bijective antipode andB⊃A a right coideal subalgebra ofA. Formally, the inclusionB⊃A defines a quotient mapG→X whereG is a quantum group andX a right homogeneousG-space. From an algebraic point of view theG-spaceX only has good properties ifA is left (or right) faithfully flat as a module overB. In the last few years many interesting examples of quantumG-spaces for concrete quantum groupsG have been constructured by Podleś, Noumi, Dijkhuizen and others (as analogs of classical compact symmetric spaces). In these examplesB consists of infinitesimal invariants of the function algebraA of the quantum group. As a consequence of a general theorem we show that in all these casesA as a left or rightB-module is faithfully flat. Moreover, the coalgebraA/AB ⁺ is cosemisimple.     

On the corestriction ofp n -symbol

LetF be a field of characteristicp. Teichmüller proved that anyp-algebra overF of indexp ⁿ and exponentp ^e is similar to a tensor product with at mostp ⁿ !(p ⁿ !−1) factors of cyclicp-algebras overF of degreep ^e . In this note we improve Teichmüller bound for two particular types ofp-algebras. LetL be a finite separable extension ofF. IfA is a cyclicp-algebra overL of degreep ^e we show that Cor _L/F A, the corestriction ofA, is similar to a tensor product with at most [L :F] factors of cyclicp-algebras overF of degreep ^e . Moreover we prove that [L :F] is the best possible bound. From this we deduce that ifA is a cyclicp-algebra overF of degreep ⁿ and exponentp ^e thenA is similar to a tensor product with at mostp ^n−e factors of cyclicp-algebras overF of degreep ^e .     

Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:Z →X which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH ¹(Z, O) →H ¹(X, EndE) is surjective. Dedicated to the memory of Professor K G Ramanathan     

Products of random varibles depending on a random walk

LetX=(X _n ) _n0 denote an irreducible random walk (ergodic in the sense of [7]) on a compact metrizable abelian groupG. In this paper we characterize completely the limit distributions of the productsY _n =X ₀...X _n . In particular we find necessary and sufficient conditions forX and/orG to imply that the products are asymptotically equidistributed in the mean, i. e. {im171-1} holds for all open,m _G -regular subsetsA ofG (m _G : normalized Haar measure).—For example ifG is monothetic and connected or ifX is asymptotically equidistributed (not merely in the mean) then the products are asymptotically equidistributed in the mean.Dedicated to Prof. Dr. L. Schmetterer on his 60^th Birthday