A subgroup H of a finite group G is called SS-supplemented in G if there exists a subgroup K of G such that HK = G and H ∩ K is S-quasinormal in K. In this paper, we characterize the finite groups in which every subgroup is SS-supplemented and the influence of SS-supplementation of some subgroups on the structure of finite groups is considered. Some recent results on SS-quasinormal subgroups and C-supplemented subgroups are strengthened and enriched.

A-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f(T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ(T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.

Historical note. A non-orientable surface of genus 2 (meaning 2 cross-caps) is popularly known as the Klein bottle. However, the term Klein surface comes from Felix Klein's book “On Riemann's Theory of Algebraic Functions and their Integrals” (1882) where he introduced such surfaces in the final chapter.

We discuss the minimum of Willmore functional of torus in a Riemannian manifold N, especially for the case that N is a product manifold. We show that when N = S2 × S1, the minimum of W(T2) is 0, and when N = R2 × S1, there exists no torus having least Willmore functional. When N = H2(−c) × S1, and x = γ × S1, the minimum of W(x) is .

It is shown that if G is a hypercentral group with all subgroups subnormal, and if the torsion subgroup of G is a π-group for some finite set π of primes, then G is nilpotent. In the case where G is not hypercentral there is a section of G that is much like one of the well-known Heineken-Mohamed groups. It is also shown that if G is a residually nilpotent group with all subgroups subnormal whose torsion subgroup satisfies the above condition then G is nilpotent.

We study the following nonlinear Dirichlet boundary value problem: where Ω is a bounded domain in ℝN(N ≥ 2) with a smooth boundary ∂Ω and g ∈ C(Ω × ℝ) is a function satisfying for all x ∈ Ω. Under appropriate assumptions, we prove the existence of infinitely many solutions when g(x, t) is not odd in t.

The purpose of this paper is to investigate the algebraic double S1-transfer, in particular the classes in the two-line of the Adams–Novikov spectral sequence which are the image of comodule primitives of the MU-homology of ℂP∞ × ℂP∞ via the algebraic double transfer. These classes are analysed by two related approaches: the first, p-locally for p ≥ 3, by using the morphism induced in MU-homology by the chromatic factorisation of the double transfer map together with the f′-invariant of Behrens (for p ≥ 5) (M. Behrens, Congruences between modular forms given by the divided β-family in homotopy theory, Geom. Topol.13(1) (2009), 319–357). The second approach (after inverting 6) uses the algebraic double transfer and the f-invariant of Laures (G. Laures, The topological q-expansion principle, Topology38(2) (1999), 387–425).

Let A = ρ2∑i,jAijθi ⊗ θj and B = ρ2∑i,jBij θi ⊗ θj be the Blaschke tensor and the Möbius second fundamental form of the immersion x. Let D = A + λB be the para-Blaschke tensor of x, where λ is a constant. If x: Mn ↦ Sn + 1(1) is an n-dimensional para-Blaschke isoparametric hypersurface in a unit sphere Sn + 1(1) and x has three distinct Blaschke eigenvalues one of which is simple or has three distinct Möbius principal curvatures one of which is simple, we obtain the full classification theorems of the hypersurface.

We give some extensions of classical results of Kellogg and Warschawski to a class of quasiconformal (q.c.) mappings. Among the other results we prove that a q.c. mapping f, between two planar domains with smooth C1,α boundaries, together with its inverse mapping f−1, is C1,α up to the boundary if and only if the Beltrami coefficient μf is uniformly α Hölder continuous (0 < α < 1).

Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

We prove that the inclusion of the space of gradient vector fields into the space of all vector fields on D2 non-vanishing in S1 is a homotopy equivalence.

We show that [z, y]−1[z, x]−1[y, x]−1[z, y][z, x][y, x] = 1 is another defining law for the variety of 3-metabelian groups.

For the Kronecker algebra, Zwara found in [14] an example of a module whose orbit closure is neither unibranch nor Cohen-Macaulay. In this paper, we explain how to extend this example to all representation-infinite algebras with a preprojective component.

In this note, we study a new Finslerian quantity Ĉ defined by the Riemannian curvature. We prove that the new Finslerian quantity is a non-Riemannian quantity for a Finsler manifold with dimension n = 3. Then we study Finsler metrics of scalar curvature. We find that the Ĉ-curvature is closely related to the flag curvature and the H-curvature. We show that Ĉ-curvature gives, a measure of the failure of a Finsler metric to be of weakly isotropic flag curvature. We also give a simple proof of the Najafi-Shen-Tayebi' theorem.

We prove a Noether–Deuring theorem for the derived category of bounded complexes of modules over a Noetherian algebra.

Let μ be a Borel measure on ℝ. The paper contains the proofs of the estimates and Here A is a subset of ℝ, f is a μ-locally integrable function, μ is the uncentred maximal operator with respect to μ and cp,q, and Cp,q are finite constants depending only on the parameters indicated. In the case when μ is the Lebesgue measure, the optimal choices for cp,q and Cp,q are determined. As an application, we present some related tight bounds for the strong maximal operator on ℝn with respect to a general product measure.

We study an (n + 1)-tensor norm αr extending to (n + 1)-fold tensor products, the classical one of Lapresté in the case n = 1. We characterise the maps of the minimal and the maximal multi-linear operator ideals related to αr in the sense of Defant and Floret (A. Defant and K. Floret, Tensor norms and operator ideals, North Holland Mathematical Studies, no. 176 (North Holland, Amsterdam, Netherlands, 1993). As an application we give a complete description of the reflexivity of the αr-tensor product (⊗j = 1n + 1 ℓuj, αr).

For an integer n ≥ 7, let Δ(n) denote the (2, 3, n)-triangle group, and let M(n) be the positive integer determined by the conditions that Δ(n) has a subgroup of index m for all m ≥ M(n), but no subgroup of index M(n) − 1. The main purpose of the paper is to obtain information (bounds, in some cases explicit values) concerning the function M(n) (cf. Theorem 1). We also show that Δ(n) is replete (i.e., has a subgroup of index m for every integer m ≥ 1) if, and only if, n is divisible by 20 or by 30 (see Theorem 2).

Let p be a prime number, Qp the field of p-adic numbers, K a finite field extension of Qp, K a fixed algebraic closure of K and Cp the completion of K with respect to the p-adic valuation. We introduce and investigate an equivalence relation on Cp, defined in terms of field extensions and metric properties of Galois orbits over K.