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.     

Restricted sumsets and a conjecture of Lev

LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev. This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.     

On galois descent for Hochschild and cyclic homology

LetG be a finite group acting by automorphisms on an algebraS over some commutative ringk. We show that if the action ofG restricted to the center ofS is Galois in the sense of [C-H-R], thenHH _*(S ^G)≊HH _* (S) ^G. An analogous result holds for cyclic homology, provided the order ofG is invertible ink. The author was supported in part by a grant from the NSF.     

A new characterization of frobenius complements

LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, ifG is a finite group andA ⊆G is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC _G(a) for 1≠a ∈A.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

Derived subgroups of fixed points

LetA be an elementary abelianq-group acting on a finiteq′-groupG. We show that ifA has rank at least 3, then properties ofC _G(a)′, 1 ≠a ∈A restrict the structure ofG′. In particular, we consider exponent, order, rank and number of generators. This author was supported by the NSF. This author was supported by CNP_q-Brazil.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Sums of lengtht in Abelian groups

LetG be an Abelian group written additively,B a finite subset ofG, and lett be a positive integer. Fort≦|B|, letB _t denote the set of sums oft distinct elements overB. Furthermore, letK be a subgroup ofG and let σ denote the canonical homomorphism σ:G→G/K. WriteB _t (modB _t) forB _tσ and writeB _t (modK) forBσ. The following addition theorem in groups is proved. LetG be an Abelian group with no 2-torsion and letB a be finite subset ofG. Ift is a positive integer such thatt<|B| then |B _t (modK)|≧|B (modK)| for any finite subgroupK ofG.     

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.     

Irreducible disconnected systems in groups

LetG be an arbitrary group with a subgroupA. The subdegrees of (A, G) are the indices [A:A ∪A ⁹] (wheregεG). Equivalent definitions of that concept are given in [IP] and [K]. IfA is not normal inG and all the subdegrees of (A, G) are finite, we attach to (A, G) the common divisor graph Γ: its vertices are the non-unit subdegrees of (A, G), and two different subdegrees are joined by an edge iff they arenot coprime. It is proved in [IP] that Γ has at most two connected components. Assume that Γ is disconnected. LetD denote the subdegree set of (A, G) and letD ₁ be the set of all the subdegrees in the component of Γ containing min(D−{1}). We proved [K, Theorem A] that ifA is stable inG (a property which holds whenA or [G:A] is finite), then the setH={g ε G| [A:A ∪A ^g ] εD ₁ ∪ {1}} is a subgroup ofG. In this case we say thatA<H<G is a disconnected system (briefly: a system). In the current paper we deal with some fundamental types of systems. A systemA<H<G is irreducible if there does not exist 1<N△G such thatAN<H andAN/N<H/N<G/N is a system. Theorem A gives restrictions on the finite nilpotent normal subgroups ofG, whenG possesses an irreducible system. In particular, ifG is finite then Fit(G) is aq-group for a certain primeq. We deal also with general systems. Corollary (4.2) gives information about the structure of a finite groupG which possesses a system. Theorem B says that for any systemA<H<G,N _G (N _G (A))=N _G (A). Theorem C and Corollary C’ generalize a result of Praeger [P, Theorem 2]. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at Tel Aviv University under the supervision of Prof. Marcel Herzog.     

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

LetC=C(C, P, k) be the coordinate ring of the affine curve obtained by removing a closed pointP from a (suitable) projective curveC over afinite fieldk. Let SL₂ (C,q) be the principal congruence subgroup of SL₂(C) andU ₂(C,q) be the subgroup generated by the all unipotent matrices in SL₂(C,q), whereq is aC-ideal. In this paper we prove that, for all but finitely manyq, the quotient SL₂(C,q)/U ₂(C,q) is a free group of finite,unbounded rank. LetC(SL₂(A)) be the congruence kernel of SL₂(A), whereA is an arithmetic Dedekind domain with only finitely many units. (e.g.A=C or ℤ) and letG be any finitely generated group. From the above (and previous results) we deduce that the profinite completion ofG,Ĝ, is a homonorphic image ofC(SL₂(A)). This is related to previous results of Lubotzky and Mel'nikov.     

Existence of a unique maximal subcoalgebra whose action is inner

LetC be a coalgebra over a fieldk. Fix an algebra map α:R →A. Introducing the notion of cleft forms, we show that, for any measuringφ:C → Hom(R, A), there is a unique maximal subcoalgebraD ofC such thatφ|_D is inner.     

On Cycles in 3-Connected Graphs

 Let G be a 3-connected graph of order n and S a subset of vertices. Denote by δ(S) the minimum degree (in G) of vertices of S. Then we prove that the circumference of G is at least min{|S|, 2δ(S)} if the degree sum of any four independent vertices of S is at least n+6. A cycle C is called S-maximum if there is no cycle C ^′ with |C ^′∩S|>|C∩S|. We also show that if ∑⁴ _i=1 d(a _i)≥n+3+|⋂⁴ _i=1 N(a _i)| for any four independent vertices a ₁, a ₂, a ₃, a ₄ in S, then G has an S-weak-dominating S-maximum cycle C, i.e. an S-maximum cycle such that every component in G−C contains at most one vertex in S. Received: March 9, 1998 Revised: January 7, 1999     

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.     

Theories of linear order

LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2^ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2^ω. (iii) IfS ₁(T) is countable then so isS _n(T) for everyn. (iv) In caseS ₁(T) is countable we find a relation between the Cantor Bendixon rank ofS ₁(T) and the Cantor Bendixon rank ofS _n(T). (v) We define a class of modelsL, and show thatS ₁(T) is finite iff the models ofT belong toL. We conclude that ifS ₁(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models. Most of the results in this paper appeared in the author’s Master of Science thesis which was prepared at the Hebrew University under the supervision of Professor H. Gaifman.     

Déformations formelles de revêtements: un principe local-global

LetC be a generically smooth, locally complete intersection curve defined over an algebraically closed fieldk of characteristicp≥0. LetG⊃ Aut _k C be a finite group of automorphisms ofC. We develop a theory ofG-equivariant deformations of the Galois coverC→C/G. We give a thorough study of the local obstructions, those localized at singular or widely ramified points, to deform equivariantly a cover. As an application, we discuss the case ofG-equivariant deformations of semistable curves.      

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

On conditional edge-connectivity of graphs

1. IntroductionIn this paper, a graph G ~ (V,E) always means a simple graph (without loops andmultiple edges) with the vertex-set V and the edge-set E. We follow [1] for graph-theoreticalterllilnology and notation not defined here.It is well known that when the underlying topology of a computer interconnectionnetwork is modeled by a graph G, the edge-connectivity A(G) of G is an important measurefor fault-tolerance of the network. However, it has many deficiencies (see [2]). MotiVatedby 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.