A subgroup H of a group G is said to be c-supplemented inG if there exists a subgroup K of G such thatHK=G and H \cap K is contained in Core_G (H). We follow Hall's ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some finite groups.

We extend splitting theorems due to Zaicev and Duan proving the following result. Let G be a locally soluble FC-hypercentral group and let A be a periodic artinian ℤG-module. If A has no finite ℤG-submodules then any extension E of A by G splits conjugately over A.

1991 Mathematics Subject Classification 20F19.

A well known result of B. Osofsky asserts that if R is a left (or right) perfect, left and right selfinjective ring thenR is quasi-Frobenius. It was subsequently conjectured by Carl Faith that every left (or right) perfect, left selfinjective ring is quasi-Frobenius. While several authors have proved the conjecture in the affirmative under some restricted chain conditions, the conjecture remains open even if R is a semiprimary, local, left selfinjective ring withJ(R)^3=0. In this paper we construct a local ring R withJ(R)^3=0 and characterize when R is artinian or selfinjective in terms of conditions on a bilinear mapping from a D-D-bimodule toD , where D is isomorphic to R/J(R). Our work shows that finding a counterexample to the Faith conjecture depends on the existence of aD -D-bimodule over a division ring D satisfying certain topological conditions.

Letu'=f\,(t,u) , u(0)=u_0 be an initial value problem with quasimonotone increasing right-hand side. We prove that if u,v are solutions such that u(t_0)\ll v(t_0) then there is a solution w withu(t_0)<w(t_0)<v(t_0) .

We describe all possible coradically graded pointed Hopf algebras of dimension p^5 (wherep is an odd prime number) over an algebraically closed field of characteristic 0.

For each m≥1, u_{m}(G) is defined to be the intersection of the normalizers of all the subnormal subgroups of defect at most m in G. An ascending chain of subgroups u_{m,i}(G) is defined by setting u_{m,i}(G)/u_{m,i−1}(G)=u_{m}(G/u_{m,i−1}(G)). If u_{m,n}(G)=G, for some integer n, the m-Wielandt length of G is the minimal of such n.

In [3], Bryce examined the structure of a finite soluble group with given m-Wielandt length, in terms of its polynilpotent structure. In this paper we extend his results to infinite soluble groups.

1991 Mathematics Subject Classification. 20E15, 20F22.

We prove the Löwner-Heinz inequality, via the Cordes inequality, for elements a,b>0 of a unital hermitian Banach *-algebra A. Letting p be a real number in the interval (0,1], the former asserts that a^p \le b^p if a \le b, a^p < b^p ifa<b , provided that the involution of A is continuous, and the latter that s(a^pb^p) \le s(ab)^p, where s(x)=r(x^*x)^{1/2} andr(x) is the spectral radius of an element x.

The structure of an inverse monoid can be determined by the complete set of Schützenberger graphs of a presentation. Necessary and sufficient conditions for a collection of inverse X-graphs to be the complete set of Schützenberger graphs of some inverse monoid presentation are established and decidability results are obtained. Conditions for a single inverse X-graph to be a Schu¨tzenberger graph for some presentation are also obtained, and both problems are restricted to the case of Clifford monoids and E-unitary inverse monoids. Decidability and undecidability results are obtained for the case of finite graphs. It is also proved that the problem of embedding a finite inverse X-graph in the Cayley graph of a group is undecidable.

Elements\alpha\in A\otimes E of the tensor product of a Banach algebra A and a Banach space E induce systems \{\psi(\alpha):\psi\in E^*\} of elements of A indexed by the dual space E^*, whose joint spectrum belongs to the second dual E^{**}. In this note we investigate when the spectrum actually lies in E\subseteq E^{**}, and extend the spectral mapping theorem P\sigma_A(\alpha)=\sigma_AP(\alpha) to polynomial mappings P:E\to F between Banach spaces. When the algebra A is commutative and the Banach space E=B is another algebra we also reach a sort of vector-valued Gelfand theory.

We define the notion of lightcone Gauss maps, lightcone pedal curves and lightcone developables of spacelike curves in Minkowski 3-space and establish the relationships between singularities of these objects and geometric invariants of curves under the action of the Lorentz group.

In this paper we give a new bound for the solutionsx of the title equation, provided that k \ge 8. This bound is polynomial in d. Moreover, under the same condition, a similar bound for the number of solutions in (x, k, y, l) is given.

It is well known that a compact embedded hypersurface of the Euclidean space without boundary is a round sphere if one of mean curvature functions is constant. In this note, we show that a compact embedded hypersurface of the Euclidean space (and other constant curvature spaces) without boundary is a round sphere if the ratio of some two mean curvature functions is constant.

1991 Mathematics Subject Classification 53C40, 53C20.

An interesting class of submanifolds of Hermitian manifolds is the class of slant submanifolds which are submanifolds with constant Wirtinger angle. In [1–4,7,8] slant submanifolds of complex projective and complex hyperbolic spaces have been investigated. In particular, it was shown that there exist many proper slant surfaces in CP^2 and inCH^2 and many proper slant minimal surfaces in C2. In contrast, in the first part of this paper we prove that there do not exist proper slant minimal surfaces inCP^2 and in CH^2. In the second part, we present a general construction procedure for obtaining the explicit expressions of such slant submanifolds. By applying this general construction procedure, we determine the explicit expressions of special slant surfaces of CP^2 and of CH^2. Consequently, we are able to completely determine the slant surface which satisfies a basic equality. Finally, we apply the construction procedure to prove that special θ-slant isometric immersions of a hyperbolic plane into a complex hyperbolic plane are not unique in general.

Let A be a noetherian connected graded ring with a balanced dualizing complex R. If A has cohomological dimension and Krull dimension 2, then

(1) R is Auslander;

(2) \rm{Cdim} M=\rm{Kdim} M for all noetherian graded A-modules M.

In particular, ifA is AS-Gorenstein of injective and Krull dimension 2, then

(3)A is Auslander-Gorenstein;

(4) A is 2-pure with a self-injective artinian quotient ring;

(5) A has a residue complex.

(1,3,4) generalize a result of Levasseur [7, 5.13] and (5) generalizes a result of Ajitabh [1, 3.12].

Given a full subcategory [Fscr ] of a category [Ascr ], the existence of left [Fscr ]-approximations (or [Fscr ]-preenvelopes) completing diagrams in a unique way is equivalent to the fact that [Fscr ] is reflective in [Ascr ], in the classical terminology of category theory.

In the first part of the paper we establish, for a rather general [Ascr ], the relationship between reflectivity and covariant finiteness of [Fscr ] in [Ascr ], and generalize Freyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories.

In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of {\rm Mod}_{R}, and has some connections with the “pure-semisimple conjecture”.

1991 Mathematics Subject Classification 18A40, 16D90, 16E70.

A module M is called a CS-module or an extending module if every submodule is essential in a direct summand of M. A ring R is called a right CS-ring or a right extending ring if R_R is a CS-module. For several types of right CS-rings it is known that either all right ideals or some large class of right ideals inherit the CS property. For example, by a result of Dung-Smith or Vanaja-Purav, a ringR is (right and left) Artinian, serial, and J(R)^2 = 0 if and only if every R-module is CS. In particular, if R is a QF-ring and J(R)^2 = 0 (hence R is serial), then everyR -module is CS. However we exhibit a finite, serial, strongly bounded QF group algebra R with J(R)^3 = 0 for which there is a principal right ideal which is a right essential extension of a CS-module and essential inR_R but not CS itself.

In this paper we study the problem of the existence on non-inner automorphisms for the class of torsion-free supersolvable groups, answering a question raised by Robinson.

1991 Mathematics Subject Classification 20F16, 20F28.

Using classification theorems ofsimple groups, we give a proof of a conjecture on factorized finite groups which is an extension of awell known theorem due to P. Hall.

Let(π_{λ}, ℋ_{λ}) be a unitary highest weight representation of the connected Lie group G and ℊ its Lie algebra. Thenℊ contains an invariant closed convex cone W_{\rm{max}} such that, for each X∈W_{\rm{max}}^0, the selfadjoint operatori·dπ_{λ}(X) is bounded from above. We show that for each suchX , the space ℋ_{λ}^{∞} of smooth vectors for the action of G on ℋ_{λ} coincides with the set𝒟^{∞}(dπ_{λ}(X)) of smooth vectors for the generally unbounded operator dπ_{λ}(X).

The univalent functions in the diagonal Besov space A_{p}, where 1<p<\infty , are characterized in terms of the distance from the boundary of a point in the image domain. Here A_{2} is the Dirichlet space. A consequence is that there exist functions in A_{p},\ p>2, for which the area of the complement of the image of the unit disc is zero.

1991 Mathematics Subject Classification 30C99, 46E35.