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.     

Onp-class groups of cyclic extensions of prime degreep of certain cyclotomic fields

Letp be an odd prime number, and letK be a cyclic extension of ℚ(ζ) of degreep, where ζ is a primitivep-th root of unity. LetC _K be thep-class group ofK, and letr _K be the minimal number of generators ofC _K ^1−σ as a module over Gal(K/ℚ(ζ)), were σ is a generator of Gal(K/ℚ(ζ)). This paper shows how likely it is forr _K = 0, 1, 2, … whenp=3, 5, or 7, and describes the obstacle to generalizing these results to regular primesp>7.     

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].     

Random generation of finite simple groups byp-regular orp-singular elements

Letp be a fixed prime and letG be a finite simple group. It is shown that two randomly chosen elements ofG of orders prime top generateG with probability tending to 1 as |G| → ∞. This answers a question of Kantor. Some related results are also established. Partially supported by a grant from the Israel Science Foundation, administered by the Israel Academy of Science and Humanities.     

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).     

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.     

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.     

Existentially complete nilpotent groups

Letn be a positive integer, letK _n denote the theory of groups nilpotent of class at mostn, and letK _n ⁺ denote the theory of torsion-free groups nilpotent of class at mostn. We show that ifn≧2 then neitherK _n norK _n ⁺ has a model companion. ForK _n we obtain the stronger result that the class of finitely generic models is disjoint from the class of infinitely generic models. We also give some other results about existentially complete nilpotent groups. Dedicated to the Memory of Abraham Robinson.     

The quotient field as a torsion-free covering module

R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom _R (G, M) there exists μ∈Hom _R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.     

On the number of copies of one hypergraph in another

Given two hypergraphsH andG, letN(G, H) denote the number of subhypergraphs ofG isomorphic toH. LetN(l, H) denote the maximum ofN(G, H), taken over allG with exactlyl edges. In [1] Noga Alon analyzes the asymptotic behaviour ofN(l, H) forH a graph. We generalize this to hypergraphs: Theorem:For a hypergraph H with fractional cover number ρ*,N(G,H).=θ(lρ*) The interesting part of this is the upper bound, which is shown to be a simple consequence of an entropy lemma of J. Shearer. In a special case which includes graphs, we also provide a different proof using a hypercontractive estimate. Research supported in part by NSF.     

Fields for which the projective schur subgroup is the whole Brauer group

LetBr(K) denote the Brauer group of a fieldK andPS(K) the projective Schur subgroup. 1. LetK be a finitely generated infinite field. ThenPS(K)=Br(K) if and only ifK is a global field. 2.|LetK be a finitely generated infinite field, and letK((t)) denote the field of formal power series int overK. ThenPS(K((t)))=Br(K((t))) if and only ifK=ℚ.     

On perfectt-shift codes in abelian groups

LetG be a finite abelian group,t a positive integer. Thet-shift sphere with centerx G is the setS _t (x)={±ix|i=1,...,t}. At-shift code is a subsetX ofG such that the setsS _t (x) (x X) have size 2t and are disjoint. Clearly, the sphere packing bound: 2t|X|+1|G| holds for anyt-shift codeX. Aperfect t-shift code is at-shift codeX with 2t|X|+1=|G|. A necessary and sufficient condition for the existence of a perfectt-shift code in a finite abelian group is known fort-1, 2. In this paper, we determine finite abelian groups in which there exists a perfectt-shift code fort=3, 4.This research was completed during the author's visit at the Institute for System Analysis, Moscow, as a Heizaemon Honda fellow of the Japan Association for Mathematical Sciences.     

The number of ideals in a quadratic field

LetK be a quadratic Geld, and letR(N) be the number of integer ideals inK with norm at most AT. Letx with conductork be the quadratic character associated withK. Then |R(N)−NL(1,x)|⩽Bk ^50/73 N ^23/73(logN)^461/146 forN ≥Ak, whereA andB are constants. ForN ≥Ak ^c,C sufficiently large, the factork ^50/73 may be replaced by (d(k))^4/73 k ^46/73. Dedicated to the memory of Professor K G Ramanathan     

Polynomial preserving maps on certain Jordan algebras

LetB andQ be associative algebras and letS be a Jordan subalgebra ofB. Letf(x ₁,…,x _m ) be a (noncommutative) multilinear polynomial such thatS is closed underf. Letα:S→Q be anf-homomorphism in the sense that it is a linear map preservingf. Under suitable conditions it is shown thatα is essentially given by a ring homomorphism. An analogous theorem forf-derivations is also proved. The proofs rest heavily on results concerning functional identities andd-freeness. The second author was partially supported by a grant from the Ministry of Science of Slovenia.     

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.     

Cyclotomic units in z p -extensions

LetK ₀ be the maximal real subfield of the field generated by thep-th root of 1 over ℚ, andK∞ be the basic Z_p-extension ofK ₀ for a fixed odd primep. LetK _n be itsn-th layer of this tower. For eachn, we denote the Sylowp-subgroup of the ideal class group ofK _n byA _n , and that ofE _n C _n byB _n , whereE _n (resp.C _n ) is the group of units (resp. cyclotomic units ofK _n . In section 2 of this paper, we describe structures of the direct and inverse limits ofB _n . The direct limit, in particular, is shown to be a direct sum of λ copies ofp-divisible groups and a finite group M, where λ is the Iwasawa λ-invariant for K∞ overK ₀. In section 3, we prove that the capitulation ofA _n inA _m is isomorphic to M form ≫n ≫ 0 by using cohomological arguments. Hence if we assume Greenberg’s conjecture (λ = 0), thenA _n is isomorphic toB _n forn ≫ 0. This paper was supported in part by a research fund for junior scholars, Korea Research Foundation The present studies were supported in part by the Basic Science Research Institute program, Ministry of Education, 1989.     

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.     

The character table of the group GL 2(Q) when extended by a certain group of order two

LetG denote either of the groupsGL ₂(q) or SL₂(q). Then θ :G →G given by θ(A) = (A ^t)^t, whereA ^t denotes the transpose of the matrixA, is an automorphism ofG. Therefore we may form the groupG.θ> which is the split extension of the groupG by the cyclic group θ of order 2. Our aim in this paper is to find the complex irreducible character table ofG. θ.     

Forests and score vectors

Thescore vector of a labeled digraph is the vector of out-degrees of its vertices. LetG be a finite labeled undirected graph without loops, and let σ(G) be the set of distinct score vectors arising from all possible orientations ofG. Let ϕ(G) be the set of subgraphs ofG which are forests of labeled trees. We display a bijection between σ(G) and ϕ(G). Supported in part by ONR Contract N00014-76-C-0366.