The Wielandt subgroup w(G) of a group G is defined to be the intersection of the normalizers of all the subnormal subgroups of G. If G is a group satisfying the minimal condition on subnormal subgroups then Wielandt [10] showed that w(G) contains every minimal normal subgroup of G, and so contains the socle of G, and, later, Robinson [6] and Roseblade [9] proved that w(G) has finite index in G.

Let

It is familiar that

(1) In the addition theorem [3, p. 363]

where

take θ = π and replace v by v−½. Since

we get

Let V be a countably generated right vector space over a division ring D. If D ≇ ℤ/2ℤ, ℤ/3ℤ, then for any γ ∈ EndD(V), there exists α ∈ AutD(V) such that γ+α, γ−α−1 ∈ AutD(V). This gives a generalization of [D. Zelinsky, Proc. Amer. Math. Soc. 5 (1954), 627–630, Theorem].

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.

Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.

We will denote the dimension of a subspace M of X by dim M and the codimension of M with respect to X by codxM or simply cod M if there is no danger of confusion. The classes of infinite dimensional and closed infinite codimensional subspaces of X will be denoted by and respectively with ℱ(X) and ℱ(X) denoting the classes of finite dimensional and of finite codimensional subspaces of X respectively. For a subspace M of X we denote the injection of M into X by and the quotient map from X onto the quotient space X/M by . Where there is no danger of confusion we will write JM and QM. The injection of X into its completion will be denoted by Jx. Letting X′ denote the continuous dual of X we remark that since X′ is isometric to ()′, these two spaces will be considered identical where convenient. The orthogonal complements of subsets M ⊂ X in X′ and K ⊂ X′ in X will be denoted by M⊥ and ⊥K respectively; M⊥X and X⊥K will be used if there is danger of confusion.

Let C be a non-empty closed convex subset of a reflexive and strictly convex Banach space E which also has a weakly continuous duality map Jφ(x) with the gauge φ. Let S and T be non-expansive mappings from C into itself such that F = F(S) ∩ F(T) ≠ ∅. Let {αn} and {βn} be sequences in (0, 1). Let {xn} be a sequence defined by where u ∈ C is a given point. Assume that the following restrictions imposed on the control sequences are satisfied:

We prove that if the order-one differential operator $S=\partial_1 + \sum_{i=2}^{n} \beta_i\partial_i + \gamma$, with $\beta_i,\gamma \in K[x_1,\ldots,x_n]$, generates a maximal left ideal of the Weyl algebra $A_n(K)$, then $S$ does not admit any Darboux differential operator in $K[x_1,\ldots,x_n]\langle \partial_2,\ldots,\partial_n\rangle $; hence in particular, the derivation $\partial_1 + \sum_{i=2}^{n} \beta_i\partial_i$ does not admit any Darboux polynomial in $K[x_1,\ldots,x_n]$. We show that the converse is true when $\beta_i \in K[x_1,x_i]$, for every $i=2,\ldots,n$. Then, we generalize to $K[x_1,\ldots,x_n]$ the classical result of Shamsuddin that characterizes the simple linear derivations of $K[x_1,x_2]$. Finally, we establish a criterion for the left ideal generated by $S$ in $A_n(K)$ to be maximal in terms of the existence of polynomial solutions of a finite system of differential polynomial equations.

The concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantised Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained by Carvalho et al. (Carvalho et al., Injective modules over down-up algebras, Glasgow Math. J. 52A (2010), 53–59) for down-up algebras.

Given an integer g ≥ 2 and a class of finite groups let N(g, ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, ) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:

A compact Klein surface X is a compact surface with a dianalytic structure. Such a surface is said to be q-hyperelliptic if it admits an involution \phi , that is an order two automorphism, such that X/ < \phi > has algebraic genus q. A Klein surface of genus 1 and k boundary components is a k-bordered torus.

By means of NEC groups, q-hyperelliptic k-bordered tori are studied and a geometrical description of their associated Teichmüller spaces is given.

Let Pn be the class of normalised polynomials of the form

of degree n which are univalent in U = {|z| < 1}. In this note we discuss the coefficients of polynomials in Pn and in some of its subclasses.

We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.

Given a number field K, we find two simple separate necessary and sufficient conditions on a given algebraic number for it to be expressible as a quotient (respectively as a difference) of two algebraic numbers conjugate over K.

We investigate the constancy of the Milnor number of one parameter deformations of holomorphic germs of functions $f:(\C^n,0) \to (\C,0)$ with isolated singularity, in terms of some Newton polyhedra associated to such germs.

When the Jacobian ideals $J(\hspace*{1.8pt}f_t) = \left\langle {\partial f_t}/{\partial x_{1}} \ldots ,{\partial f_t}/{\partial x_{n}}\right\rangle $ of a deformation $f_t(x) = f(x)+ \sum_{s=1}^{\ell}\delta_s(t)g_s(x)$ are non-degenerate on some fixed Newton polyhedron $\Gamma_+$, we show that this family has constant Milnor number for small values of $t$, if and only if all germs $g_s$ have non-decreasing $\Gamma$-order with respect to $f$. As a consequence of these results we give a positive answer to Zariski's question for Milnor constant families satisfying a non-degeneracy condition on the Jacobian ideals.

Bellman [1], [2, p. 116] proved that, if all solutions of the equation

are in L2, ∞) and b(t) is bounded, then all solutions of

are also in L2(a, ∞). The purpose of this paper is to present conditions on the function f that guarantee that all solutions of

be in the class L2(a, ∞) whenever all solutions of the equation

have this property. It is assumed that r(t) >0, r and qare continuous on a half line (a, ∞) and f is continuous. Actually the continuity assumptions may be weakened to local integrability and L2 (a, ∞) may be replaced by Lp(a, ∞) for any p > 1.

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).

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.

for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

Many of the techniques and notions used to study various important theorems in locally convex spaces are not effective for general linear topological spaces. In [4], a study is made of notionsin general linear topological spaces which can be used to replace barrelled, bornological, and quasi-barrelled spaces. The present paper contains a parallel study in the context of semiconvex spaces.

Existence theorems for the equation F(x)=\varphi (x) are proved when F is a function with “good” surjective properties and \varphi satisfies certain compactness conditions on countable subsets of the space. Also results for certain homotopic perturbations of the equation are obtained. The results lead to various fixed point theorems of Darbo type for F=id, but they are also applicable if F acts between different spaces. Also the inclusions F(x)\in \varphi (x) (resp. F(x)\subseteq \varphi (x)) for multivalued functions \varphi (resp. F and \varphi) are studied. There are some connections with the theory of 0-epi maps.