Generalized Frobenius groups

A pair (G. K) in whichG is a finite group andK◃G, 1<K<G, is said to satisfy (F2) if |C _G (x)|=|C _G/K (xK)| for allx∈G/K. First we survey all the examples known to us of such pairs in whichG is neither ap-group nor a Frobenius group with Frobenius kernelK. Then we show that under certain restrictions there are, essentially, all the possible examples.     

Semilinear elliptic equations with uniform blow-up on the boundary

We prove the existence and the uniqueness of a solutionu of−Lu+h|u| ^α-1u=f in some open domain ℝ^d, whereL is a strongly elliptic operator,f a nonnegative function, and α>1, under the assumption that ∂G is aC ² compact hypersurface, lim _x→∂G (dist(x, ∂G))^2α/(α-1) f(x)=0, and lim _x→∂G u(x)=∞.     

On the boundedness of solutions of nonlinear differential equations in Hilbert spaces

LetH be a complex Hilbert space and letB be the space of all bounded linear operators fromH intoH with the strong operator topology. We will give a boundedness result for the solutions of the differential equationx′=A(t)x+f(t,x) whereA: I=[t ₀, ∞)→B is continuous,f: I×H→H is also continuous and for every bounded setS⊂I×H there exists a constantM(S)>0 such that |f(t,x)−f(t,y)|≤M(S)|x−y|,(t,x), (t,y)∈S.

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Sunto SiaH uno spazio di Hilbert complesso e siaB lo spazio degli operatori lineari limitati daH inH, con la topologia forte. In questo lavoro si prova un risultato di limitatezza per le soluzioni dell'equazione differenzialex′=A(t)x+f(t,x), doveA: I=[t ₀, ∞)→B è continua,f: I×H→H è continua e per ogni insieme limitatoS⊂I×H esiste una costanteM(S)>0 tale che |f(t,x)−f(t,y)|≤M(S)|x−y| per ogni(t,x), (t,y)∈S.

     

On the local Artin conductor f<Subscript>Artin</Subscript> (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK _k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren _G is the break in the upper ramification filtration ofG = Gal(E/K) defined by . Next, we study the basic properties of the idealf(E/K) inO _k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin charactera _G : G → ℂ ofG := Gal(E/K) and Artin representationsA _g G → G →GL(V) corresponding toa _G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5) where Χ_gr : G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then where Gal(E^ker(ρ)/E^ker(ρ)•) is any maximal abelian normal subgroup of Gal(E^ker(ρ)/K) containing Gal(E^ker(ρ) /K)′, and the break n_G/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A _G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V _n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^• → ℂ^Χ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_δ} = ω =(G/N) and where δ runs over R((G/N)^•/(G/N)), a fixed given complete system of representatives of (G/N)^•/(G/N), by declaring that ω₁ ∼ ω₂ if and only if ω₁ = ω _2,δ for some δ ∈ R((G/N)^•/(G/N)). Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.     

On polynomiality of the method of analytic centers for fractional problems

We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ ⁿ is a closed and bounded convex domain,K ⊂ ℝ ^m is a closed convex cone andA(x):G → ℝ ⁿ ,B(x):G→K are regular enough (say, affine) mappings. This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences     

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.     

On the Modular Version of the Gluck-Wolf Theorem

Let N be a normal subgroup of a finite group G. Let ϕ be an irreducible Brauer character of N. Assume π is a set of primes and χ(1)/ϕ(1) is a π′-number of any χ∈IBr _p (G/ϕ). If p∤|G:N|, and N is p-solvable, then G/N has an abelian-by-metabelian Hall-π subgroup; If p∉π then G/N has a metabelian Hall-π subgroup. Received February 22, 2000, Accepted May 9, 2001     

Hyperidentities of lattices and semilattices

Following W. Taylor we define a hyperidentity ∈ to be formally the same as an identity (e.g.,F(G(x, y, z), G(x, y, z))=G(x, y, z)). However, a varietyV is said to satisfy a hyperidentity ∈, if whenever the operation symbols of ∈ are replaced by any choice of polynomials (appropriate forV) of the same arity as the corresponding operation symbols of ∈, then the resulting identity holds inV in the usual sense. For example, if a varietyV of type <2,2> with operation symbols ∨ and ∧ satisfies the hyperidentity given above, then substituting the polynomial (x∨y)∨z for the symbolG, and the polynomialx∧y forF, we see thatV must in particular satisfy the identity ((x∨y)∨z)∧((x∨y)∨z)=((x∨y)∨z). The set of all hyperidentities satisfied by a varietyV, will be denoted byH(V). We shall letH ^m (V) be the set of all hyperidentities hoiding inV with operation symbols of arity at mostm, andH _n (V) will denote the set of all hyperidentities ofV with at mostn distinct variables. In this paper we shall show that ifV is a nontrivial variety of lattices or the variety of all semilattices, then for any integersm andn, there exists a hyperidentity ∈ such that ∈ holds inV, and ∈ is not a consequence ofH ^m (V)∪H _n (V). From this it is deduced that the hyperidentities ofV are not finitely based, partly soling a problem of Taylor [7, Problem 3]. The research of the author was supported by NSERC of Canada. Presented by W. Taylor.     

On finite groups whose derived subgroup has bounded rank

Let G be a finite group with derived subgroup of rank r. We prove that |G: Z ₂(G)| ≤ |G′|^2r . Motivated by the results of I. M. Isaacs in [5] we show that if G is capable then |G: Z(G)| ≤ |G′|^4r . This answers a question of L. Pyber. We prove that if G is a capable p-group then the rank of G/Z(G) is bounded above in terms of the rank of G′.     

Generalized frobenius groups. II

A pair (G, K) in whichG is a finite group andK a normal nontrivial proper subgroup ofG is said to be an F2-pair (a Frobenius type pair) if |C _G (x)|=|C _G/K (xK)| for allx∈G\K. A theorem of Camina asserts that in this case eitherK orG/K is ap-group or elseG is a Frobenius group with Frobenius kernelK. The structure ofG will be described here under certain assumptions on the Sylowp-subgroups ofG. This author’s research was partially supported by the Technion V.P.R. fund — E.L.J. Bishop research fund. This author’s research was partially supported by the MPI fund.     

On the regularity of the minima of quasiconvex integrals

We prove in this paper theC ^∞ regularity for a “very strict” local minimum of classC _loc ^ρ , ρ>3, of functionals with genuine degenerate quasiconvex integrand depending on a vector-valued function u. Such a minimum satisfies the condition: for all x∈Ω, there exists a neighbourhoodK(x) ofx in Ω andC ₁ (x)>0,C ₂ (x)>0,1≥ε(x)>0, such that for all real ϕ∈c ₀ ^∞ (K). This work is supported by the National Natural Science Foundation of China and the Fok Ying Tung Education Foundation.     

Groups with an almost regular involution

An involution j of a group G is said to be almost perfect in G if any two involutions in j^G whose product has infinite order are conjugated by a suitable involution in j^G. Let G contain an almost perfect involution j and |C_G(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |C_G(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite index in G. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 360–368, May–June, 2007.     

Asymptotic behavior of the solution to the two-dimensional stationary problem of flow past a body far from it

In the exterior domain Ω⊂ℝ² we consider the two-dimensional Navier-stokes system Δu-▽p=(u,▽)u, div u=0 whose solution possesses a finite Dirichlet integral and satisfies the condition lim_|x|→∞ u(x)=(1, 0). For this solution, we establish the estimate |u(x)−(1, 0)|≤c|x| ^−α, where α>1/4. This estimate implies an asymptotic expression for the solution indicating the presence of a track behind the body. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 246–253, February, 1999.     

Oscillation of nonlinear vector differential equations

Summary Oscillation criteria are obtained for vector partial differential equations of the type Δv+b(x, v)v=0, x∈G, v∈E^m, where G is an exterior domain in E_n, and b is a continuous nonnegative valued function in G × E^m. A solution v: G→E^m is called h-oscillatory in G whenever the scalar product [v(x), h] (|h|=1) has zeros x in G with |x| arbitrarily large. It is shown that the spherical mean of [v(x), h] over a hypersphere of radius r in Eⁿ satisfies a nonlinear ordinary differential inequality. As a consequence, the main theorems give sufficient conditions on b(x, t), depending upon the dimension n, for all solutions v to be h-oscillatory in G. Entrata in Redazione il 26 giugno 1975.     

Lower bounds for the number of conjugacy classes in finite solvable groups

We prove first that if G is a finite solvable group of derived length d ≥ 2, then k(G) > |G|^1/(2^d−1), where k(G) is the number of conjugacy classes in G. Next, a growth assumption on the sequence [G⁽ⁱ⁾: G⁽ⁱ⁺¹⁾] ₁ ^d−1 , where G⁽ⁱ⁾ is theith derived group, leads to a |G|^1/(2d−1) lower bound for k(G), from which we derive a |G|^c/log ₂log₂|G| lower bound, independent of d(G). Finally, “almost logarithmic” lower bounds are found for solvable groups with a nilpotent maximal subgroup, and for all Frobenius groups, solvable or not.     

Movement of Intransitive Permutation Groups Having Maximum Degree

Let G be a permutation group on a set Ω with no fixed points in,and m be a positive integer.Then the movement of G is defined as move(G):=sup Γ {|Γg\Γ| | g ∈ G}.It was shown by Praeger that if move(G) = m,then |Ω| 3m + t-1,where t is the number of G-orbits on.In this paper,all intransitive permutation groups with degree 3m+t-1 which have maximum bound are classified.Indeed,a positive answer to her question that whether the upper bound |Ω| = 3m + t-1 for |Ω| is sharp for every t > 1 is given.     

Optimal properties of contraction semigroups in banach spaces

Suppose that(T _t )_t>0 is aC ₀ semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ ⁻¹(Jx)={x}, whereJ is the duality mapping fromX toP(X ^*). If |<T _t x,f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible. If the hypothesisJ ⁻¹(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ ⁻¹(Jx) is weakly compact, then if |<T _t x, f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1, there existsy∈J ⁻¹(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ₁ orL ₁.     

Negative result in pointwise 3-convex polynomial approximation

Let Δ³ be the set of functions three times continuously differentiable on [−1, 1] and such that f″′(x) ≥ 0, x ∈ [−1, 1]. We prove that, for any n ∈ ℕ and r ≥ 5, there exists a function f ∈ C ^r [−1, 1] ⋂ Δ³ [−1, 1] such that ∥f ^(r)∥ _{C[−1, 1]} ≤ 1 and, for an arbitrary algebraic polynomial P ∈ Δ³ [−1, 1], there exists x such that

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.