Let G be a group and let H be a subgroup of G. The automizer AutG(H) of H in G is defined as the group of automorphisms of H induced by conjugation of elements of NG(H). Thus AutG(H)≅NG(H)/CG(H), and we obviously have $$\tfrm{In(}H\tfrm{)≤Aut}_{G}\tfrm{(}H\tfrm{)≤Aut(}H\tfrm{).}$$ We call AutG(H) large if AutG(H)=Aut(H) and small if AutG(H)=In(H).

We denote by $\rm{Aut}_{sn}(G)$ the set of all automorphisms that fix every subnormal subgroup of $G$ setwise. In their paper [5], Franciosi and de Giovanni began the study of $\rm{Aut}_{sn}(G)$. Other authors have also considered the structure of $\rm{Aut}_{sn}(G)$ under various restrictions on the structure of $G$ (Robinson [11], Cossey [2], Dalle Molle [4]). The inner automorphisms in $\rm{Aut}_{sn}(G)$ are precisely the inner automorphisms induced by elements of $\omega(G)$, the Wielandt subgroup of $G$. Recall that the Wielandt subgroup of a group $G$ is the set of all elements of $G$ that normalise each subnormal subgroup of $G$ and that $\zeta(G)$, the centre of $G$, is contained in $\omega;(G)$. Thus $\rm{Aut}_{sn}(G)\cap;\rm{Inn}(G)$ is isomorphic to $\omega(G)/\zeta(G)$ and some of the results obtained indicate that the structure of $\rm{Aut}_{sn}(G)$ is controlled by the structure of $\omega(G)/\zeta(G)$; for example, Robinson [11, Corollary 3] shows that, for a finite group $G,\rm{Aut}_{sn}(G)$ is insoluble if and only if $\omega(G)$ is insoluble. We shall prove a result of a similar nature here. One of the main results (Theorem B) of Franciosi and de Giovanni [5] is that, for a polycyclic group $G$, $\rm{Aut}_{sn}(G)$ is either finite or abelian. We shall show that $\rm{Aut}_{sn}(G)$ can indeed be infinite, but only if $\omega(G)/\zeta(G)$ is infinite.

We establish an estimate for the measure of non-compactness of an interpolated operator acting from a J-space into a K-space. Our result refers to general Banach N-tuples. We also derive estimates for entropy numbers if some of the N-tuples reduce to a single Banach space.

We show that if R is a ring such that each minimal left ideal is essential in a (direct) summand of RR, then the dual of each simple right R-module is simple if and only if R is semiperfect with Soc(RR)=Soc(RR) and Soc(Re) is simple and essential for every local idempotent e of R. We also show that R is left CS and right Kasch if and only if R is a semiperfect left continuous ring with Soc(RR)⊆eRR. As a particular case of both results we obtain that R is a ring such that every (essential) closure of a minimal left ideal is summand (R is then said to be left strongly min-CS) and the dual of each simple right R-module is simple if and only if R is a semiperfect left continuous ring with Soc(RR)=Soc(RR)⊆eRR. Moreover, in this case R is also left Kasch, Soc(eR)≠0 for every local idempotent e of R, and R admits a (Nakayama) permutation of a basic set of primitive idempotents. As a consequence of this result we characterise left PF rings in terms of simple modules over the 2×2 matrix ring by showing that R is left PF if and only if M2(R) is a left strongly min-CS ring such that the dual of every simple right module is simple.

We prove an $n$-tuple analogue of the Putnam area inequality for the spectrum of a single $p$-hyponormal operator.

We present an algorithm for computing the Green function of the weighted biharmonic operator Δ|P′|−2Δ on the unit disc (with Dirichlet boundary conditions) for rational functions P. As an application, we show that if P is a Blaschke product with two zeros α1, α2 the Green function is positive if and only if |(α1−α2)/(1−{\bar α}1α2)|≤{2 \over 7}{\sqrt 10}, and also obtain an explicit formula for the Green function of the operator Δ|G|−2Δ, where G is the canonical zero-divisor of a finite zero set on the Bergman space.

A group G is said to be a minimal non-FC group, if G contains an infinite conjugacy class, while every proper subgroup of G merely has finite conjugacy classes. The structure of imperfect minimal non-FC groups is quite well-understood. These groups are in particular locally finite. At the other end of the spectrum, a perfect locally finite minimal non-FC group must be a p-group. And it has been an open question for quite a while now, whether such groups exist or not.

We prove that the conditions R·R=0 and R·S=0 are equivalent for hypersurfaces of a 5-dimensional semi-Riemannian space form N5(c). This solves a problem by P.J. Ryan in the case of hypersurfaces of dimension 4 in semi-Riemannian space forms.

Let πc(x) be the number of the integers n≤x such that [nc] is prime. We shall prove that $$π_{c}\tfrm{(}x\tfrm{)&2_Gt;}{x\over \log x}$$ for 1<c<45/38. This improves the former range 1<c<13/11.

By a quasi-permutation matrix we mean a square matrix over the complex field [Copf ] with non-negative integral trace. Thus every permutation matrix over [Copf ] is a quasi-permutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational field ℚ, and let c(G) be the minimal degree of a faithful representation of G by complex quasi-permutation matrices. See [1].

We give a method allowing the generalization of the description of trace spaces of certain classes of holomorphic functions on Carleson sequences to finite unions of Carleson sequences. We apply the result to different classes of spaces of holomorphic functions such as Hardy classes and Bergman type spaces.

A subgroup M of an infinite group G is said to be nearly maximal if it is a maximal element of the set of all subgroups of G having infinite index; i.e. if the index |G:M| is infinite but every subgroup of G properly containing M has finite index in G. The near Frattini subgroup ψ(G) of an infinite group G can now be defined as the intersection of all nearly maximal subgroups of G, with the stipulation that ψ(G)=G if G has no nearly maximal subgroups. These concepts have been introduced by Riles [5]. It was later proved by Lennox and Robinson [4] that a finitely generated soluble-by-finite group G is infinite-by-nilpotent if and only if all its nearly maximal subgroups are normal. It follows that in the class of finitely generated soluble-by-finite groups the property of being finite-by-nilpotent is inherited from the near Frattini factor group G/ψ(G) to the group G itself. In the study of ordinary Frattini properties of infinite groups, some analogies exist between the behaviour of finitely generated soluble groups and soluble minimax residually finite groups (see for instance [6] and [7]). This fact could suggest that a result corresponding to that of Lennox and Robinson also holds for soluble residually finite minimax groups. Unfortunately in this case the property of being finite-by-nilpotent cannot be detected from the behaviour of nearly maximal subgroups, this phenomenon depending on the fact that infinite soluble residually finite minimax groups may be poor of such subgroups.

We give the first examples of contact metric spaces which are weakly locally $\varphi$-symmetric (that is, $\nabla R=0$ on horizontal vectors), but not strongly (that is, not all reflections with respect to the characteristic lines are local isometries). These examples are three-dimensional non-unimodular Lie groups with a left-invariant contact metric structure. We exhibit additional symmetries on these spaces.

1991 Mathematics Subject Classification: 53B20, 53C15, 53C25.

If A is an algebra and ϑ is a congruence on A then A is said to be ϑ-coherent provided that, for every subalgebra B of A, if B contains some ϑ-class then B is a union of ϑ-classes. An algebra A is said to be congruence coherent if it is ϑ-coherent for every ϑ∈>ConA. This notion was investigated by Beazer [2] in the context of de Morgan algebras. Specifically, he showed that a de Morgan algebra is congruence coherent if and only if it is boolean, or simple, or the 4-element de Morgan chain. He also showed that if an algebra in the Berman class K1,1 of Ockham algebras is congruence coherent then it is necessarily a de Morgan algebra; and that a p-algebra is congruence coherent if and only if it is boolean. This notion has also been considered in the context of distributive double p-algebras by Adams, Atallah and Beazer [1] who showed that particular examples of congruence coherent double p-algebras are those that are congruence regular (in the sense that if two congruences have a class in common then they coincide). In this paperNATO Collaborative Research Grant 960153 is gratefully acknowledged. we extend the results of Beazer to the class of double MS-algebras.

A finite group G can be represented as a group of automorphisms of a compact Riemann surface, that is, G acts on a Riemann surface. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts (possibly reversing orientation).

The nonabelian tensor square G[otimes ] G of a group G is generated by the symbols g[otimes ] h, g,h ∈ G, subject to the relations $$gg\prime\otimesh=(^gg\prime\otimes^gh)(g\otimesh) and g\otimeshh\prime-(g\otimesh)(^hg\otimes^hh\prime),$$ for all $g,g\prime,h,h\prime \in G< / f>, where $^gg\prime=gg\primeg^{−1}$. The nonabelian tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J.-L. Loday in [4] and [5], extending ideas of J.H.C. Whitehead in [10]. The topic of this paper is the classification of 2-generator 2-groups of class two up to isomorphism and the determination of nonabelian tensor squares for these groups.

Using the Hilbert Uniqueness Method, we study the problem of exact controllability in Neumann boundary conditions for problems of transmission of the wave equation. We prove that this system is exactly controllable for all initial states in L2(Ω)×(H1(Ω))′.

This paper gives examples to show that a polar operator is not necessarily AC and an AC operator is not necessarily polar.

Let $X$ be a Banach space and let $B(X)$ denote the space of bounded operators on $X$. Two elements $S,T\inB(X)$ are isometrically equivalent if there exists an invertible isometry $V$ such that $TV=VS$. If $X$ is a Hilbert space, then $V$ is a unitary operator and $S$ and $T$ are said to be unitarily equivalent.

The notion of quadratic congruences was introduced in the recently published paper [A. Balog, H. Darmon and K. Ono, Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions, in Analytic Number Theory, Vol. 1, Progr. Math.138 (Birkhäuser, Boston, 1996), 105–128.]. In this note we present different, somewhat more conceptual proofs of several results from that paper. Our method allows us to refine the notion and to generalize the results quoted. Here we deal only with the quadratic congruences for Cohen–Eisenstein series. Similar phenomena exist for cusp forms of half-integral weight as well; however, as one would expect, in the case of Eisenstein series the argument is much simpler. In particular, we do not make use of techniques other than p-adic Mazur measure, whereas the consideration of cusp forms of half-integral weight involves a much more sophisticated construction. Moreover, in the case of Cohen–Eisenstein series we are able to obtain a full and exhaustive result. For these reasons we present the argument here.