Using the definition of regular p-group given by M. Hall [1], a new class of finite groups called regular-nilpotent has been defined. The action of these groups as automorphisms of compact Riemann surfaces has been investigated. It is proved that a necessary and sufficient condition for a Fuchsian group to cover a regular-nilpotent group is that its orbit genus be zero and its periods satisfy the least common multiple condition, first defined by Harvey [2] and Maclachlan [4].

For an arbitrary $K$-algebra $R$, an $R$, $K$-bimodule $M$ is algebraically reflexive if the only $K$-endomorphisms of $M$ leaving invariant every $R$-submodule of $M$ are the scalar multiplications by elements of $R$. Hadwin has shown for an infinite field $K$ and $R = K[x]$ that $R$ is reflexive as an $R$, $K$-bimodule. This paper provides a generalisation by giving a skew polynomial version of his result.

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. We show that B(X) is finite if and only if no proper, complemented subspace of X is isomorphic to X, and we show that B(X) is properly infinite if and only if X contains a complemented subspace isomorphic to X[oplus ]X. We apply these characterizations to find Banach spaces X1, X2, and X3 such that B(X1) is finite, B(X2) is infinite, but not properly infinite, and B(X3) is properly infinite. Moreover, we prove that every unital, properly infinite ring has a continued bisection of the identity, and we give examples of Banach spaces D1 and D2 such that B(D1) and B(D2) are infinite without being properly infinite, B(D1) has a continued bisection of the identity, and B(D2) has no continued bisection of the identity. Finally, we exhibit a unital $C^\ast$-algebra which is finite and has a continued bisection of the identity.

We consider associative algebras $\Lambda$ over a field provided with a direct sum decomposition of a two-sided ideal $M$ and a sub-algebra $A$ – examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of $\Lambda$. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.

Finite groups in which the Frattini subgroup of each proper normal subgroup is trivial, while the group itself has a nontrivial Frattini subgroup, are investigated. A direct result of this study leads to a classification of finite groups in which the Frattini subgroup of each proper subgroup is trivial, while the group itself has a nontrivial Frattini subgroup.

Recently, Ballester-Bolinches [1 and 2], Pedraza-Aguilera [2] and Perez-Ramos [2] have studied circumstances under which certain injectors and projectors, which are always pronormal, must be normally embedded. In this note we give a scheme for describing a minimal counterexample to a conjecture of the form: a subnormally embedded subgroup with properties $\alpha_1$, $\alpha_2,{\ldots\,},\alpha_{n}$ is normally embedded, where $\alpha_1$, $\alpha_2,{\ldots\,},\alpha_{n}$ satisfy certain conditions. We then show contradictions in certain cases involving finite solvable groups.

Extending results and methods of Thomson and Trent, we prove the existence of non trivial quasi-invariant subspaces for subnormal families of unbounded operators having sufficiently rich domains. In some special cases, proper invariant subspaces are obtained.

Let $\rho:G\hookrightarrow {\rm GL}(n, {\bb F})$ be a representation of a finite group over the field ${\bb F}, V = {\bb F}^n$ the corresponding $G$-module, and ${\bb F}[V\hspace*{1.5pt}]$ the algebra of polynomial functions on $V$. The action of $G$ on $V$ extends to ${\bb F}[V\hspace*{1.5pt}]$, and ${\bb F}[V\hspace*{1.5pt}]^G$, respectively ${\bb F}[V\hspace*{1.5pt}]_G$, denotes the ring of invariants, respectively coinvariants. The theorem of Steinberg referred to in the title says that when ${\bb F} = {\bb C}$, dim$_{\bb C}$ (Tot$({\bb C}[V\hspace*{1.5pt}]_G))=|G|$ if and only if $G$ is a complex reflection group. Here Tot$({\bb F}[V\hspace*{1.5pt}]_G)$ denotes the direct sum of all the homogeneous components of the graded algebra ${\bb F}[V\hspace*{1.5pt}]_G$ and $|G|$ is the order of $G$. Chevalley's theorem tells us that the ring of invariants of a complex pseudoreflection representation $G\hookrightarrow {\rm GL}(n, {\bb C})$ is polynomial algebra, and the theorem of Shephard and Todd yields the converse. Combining these results gives: dim$_{\bb F}$(Tot$({\bb C}[V\hspace*{1.5pt}]_G) = |G|$ if and only if ${\bb C}[V\hspace*{1.5pt}]^G$ is a polynomial algebra. The purpose of this note is to show that the two conditions

(i) dim$_{\bb F}$(Tot(${\bb F}[V\hspace*{1.5pt}]_G))\,{=}\,|G|$,

(ii) ${\bb F}[V\hspace*{1.5pt}]^G$ is a polynomial algebra

are equivalent regardless of the ground field; in particular in the modular case.

Two subgroups $M$ and $S$ of a group $G$ are said to permute, or $M$permutes with$S$, if $MS = SM$. Furthermore, $M$ is a permutable subgroup of $G$ if $M$ permutes with every subgroup of $G$. In this article, we provide necessary and sufficient conditions for a subgroup of $G\times H$, whose intersections with the direct factors are normal, to be a permutable subgroup.

From the classical differential equation of Jacobi fields, one naturally defines the Jacobi operator of a Riemannian manifold with respect to any tangent vector. A straightforward computation shows that any real, complex and quaternionic space forms satisfy that any two Jacobi operators commute. In this way, we classify the real hypersurfaces in quaternionic projective spaces all of whose tangent Jacobi operators commute.

We prove that the perturbation class of the upper semi-Fredholm operators from $X$ into $Y$ is the class of the strictly singular operators, whenever $X$ is separable and $Y$ contains a complemented copy of $C[0, 1]$. We also prove that the perturbation class of the lower semi-Fredholm operators from $X$ into $Y$ is the class of the strictly cosingular operators, whenever $X$ contains a complemented copy of $\ell_1$ and $Y$ is separable. We can remove the separability requirements by taking suitable spaces instead of $C[0, 1]$ or $\ell_1$.

Let $J_\mu$ denote the Bessel function of order $\mu$. The functions $x^{-\alpha-1} J_{\alpha+2n+1}(x), n=0,1,2,\ldots,$ form an orthogonal system in the space $L^2((0,\infty), x^{2\alpha+1}dx)$ when $\alpha >{-}1$. In this paper we prove that the Fourier series associated to this system is of restricted weak type for the endpoints of the interval of mean convergence, while it is not of weak type if $\alpha\,{\ge}\,0$.

Let $\alpha$ be an irrational number. We determine the Hausdorff dimension of sets of real numbers which are close to infinitely many elements of the sequence $(\{n\alpha\})_{n\,{\ge}\,1}$.

Let $A$ be a locally finite, N-graded, noetherian algebra with a balanced dualizing complex. If $A$ is a Hopf algebra, then $A$ has finite injective dimension.

We derive variational formulae for natural first order energy functionals and obtain criteria for the stability of isometric immersions. This generalizes known results for the classical energy, the $p$-energy and the exponential energy.

Page 188 in Ramanujan's lost notebook is devoted to a certain class of infinite series connected with Euler's pentagonal number theorem. These series are represented in terms of Ramanujan's famous Eisenstein series $P, Q$, and $R$. The purpose of this paper is to prove all the formulas on page 188 and to show that one of them leads to an interesting, new recurrence formula for $\sigma (n)$, the sum of the positive divisors of $n$.

In this paper, a dual version of the Mountain Pass Theorem and the Generalized Mountain Pass Theorem are extended to functions that are locally Lipschitz only. An application involving elliptic hemivariational inequalities is next examined.

We show that surfaces such that the natural projections of the unit normal bundles are harmonic morphisms are composed of minimal points and totally umbilical points. As its application, we find a harmonic map from the torus to the complex quadric in $CP^3$ such that the projection map of the associated sphere bundle constructed by Gudmundsson is not a harmonic morphism. This contrasts sharply with the situation for holomorphic maps. We also establish sufficient conditions for reducing the codimension of an isometric immersion with conformal second fundamental form.

For $p\ge 2$ we introduce the notion of an almost $p$-structure on vector-bundles which generalizes the notion of an almost-complex structure and investigate the existence of almost $p$-structures on spheres and complex projective spaces.

For a subset $\Lambda$ of the dual group of a compact metrizable abelian group, we introduce the type I-, II-, and III-$\Lambda$-Riemann–Lebesgue property of a Banach space. As an application we use these properties to characterize Rajchman sets.