Tensor analogues of right 2-Engel elements in groups were introduced by D. P. Biddle and L.-C. Kappe. We investigate the properties of right 2-Engel tensor elements and introduce the concept of $2_{\otimes}$-Engel margin. With the help of these results we describe the structure of $2_{\otimes}$-Engel groups. In particular, we prove a tensor version of Levi's theorem for 2-Engel groups and determine tensor squares of two-generator $2_{\otimes}$-Engel $p$-groups.

Given any finite group G, we wish to determine a relationship between the highest power of a prime p dividing the order of G, denoted by |G|p, and |A(G)|p, where A(G) is the automorphism group of G. It was shown by Herstein and Adney [8] that |A(G)|p ≧ p whenever |G|p= ≧P2. Later Scott [16] showed that A(G)p≧P2. For the special case where G is abelian, Hilton [9] proved that Adney [1] showed that this result holds if a Sylow p-subgroup of G is abelian, and gave an example where |G|p= p4 and |A(G)|P =p2. We are able to show in Theorem 4.5 that, if |G|p = ≧ p5, then |A(G)| = ≧ p3.

In this paper we shall be concerned with the set Mn(B) of n x n matrices whose elements belong to a given Boolean algebra B(≤, ∪,∩,').

It is well known that Mn(B) also forms a Boolean algebra with respect to the partial ordering ≼ defined by

in which union (⋎), intersection (⋏) and complementation (*) are given by

We show that PQε-projectivity of two Riemannian metrics introduced in [15] (P. J. Topalov, Geodesic compatibility and integrability of geodesic flows, J. Math. Phys.44(2) (2003), 913–929.) implies affine equivalence of the metrics unless ε ∈ {0,−1,−3,−5,−7,. . .}. Moreover, we show that for ε=0, PQε-projectivity implies projective equivalence.

Given a group G and a positive integer k, let vk(G) denote the number of conjugacy classes of subgroups of G which are not subnormal of defect at most k. Groups G such that vkG) < ∝ for some k are considered in Section 2 of [1], and Theorem 2.4 of that paper states that an infinite group G for which vk(G) < ∝ (for some k) is nilpotent provided only that all chief factors of G are locally (soluble or finite). Now it is easy to see that a group G whose chief factors are of this type is locally graded, that is, every nontrivial, finitely generated subgroup F of G has a nontrivial finite image (since there is a chief factor H/K of G such that F is contained in H but not in K). On the other hand, every (locally) free group is locally graded and so there is in general no restriction on the chief factors of such groups. The class of locally graded groups is a suitable class to consider if one wishes to do no more than exclude the occurrence of finitely generated, infinite simple groups and, in particular, Tarski p-groups. As pointed out in [1], Ivanov and Ol'shanskiĭ have constructed (finitely generated) infinite simple groups all of whose proper nontrivial subgroups are conjugate; clearly a group G with this property satisfies v1(G) = l. The purpose of this note is to provide the following generalization of the above-mentioned theorem from [1].

Little is known about the generators system of the higher dimensional Picard modular groups. In this paper, we prove that the higher dimensional Eisenstein–Picard modular group PU(3, 1;ℤ[ω3]) in three complex dimensions can be generated by four given transformations.

It is shown that for the Laplacian masa in the free group factors, the orthocomplement of the associated Jones' projection is an infinite direct sum of coarse bimodules.

Let Mn be the C*-algebra of n × n complex matrices. If A is a C*-algebra, let Mn(A) denote the C*-algebra of n × nmatrices a = [aij] with entries in A. For a linear map between C*-algebras, we define the multiplicity map by A linear map Ø is said to be completely bounded if Let B(A, B), CB(A, B) denote the Banach space of bounded linear maps, the set of completely bounded maps from A to B, respectively.

We compute the Grothendieck and Picard groups of a smooth toric DM stack by using a suitable category of graded modules over a polynomial ring. The polynomial ring with a suitable grading and suitable irrelevant ideal functions is a homogeneous coordinate ring for the stack.

A polynomial P ∈ (kE, F) is left ℓ1-factorable if there are a polynomial Q ∈ (kE, ℓ1) and an operator L ∈ (ℓ1, F) such that P = L ○ Q. We characterise the Radon–Nikodým property by the left ℓ1-factorisation of polynomials on L1(μ). We study the left ℓ1-factorisation of nuclear, compact and Pietsch integral polynomials. For Pietsch integral polynomials, we introduce the left integral ℓ1-factorisation property, obtaining a second polynomial characterisation of the Radon–Nikodým property and showing that it plays a role somehow comparable, in this setting, to nuclearity of operators. A characterisation of 1-spaces is also given in terms of the left compact ℓ1-factorisation of polynomials.

The origin of the theory of averaging operators is explained in [1]. The theory has been developed on spaces of continuous functions that vanish at infinity by Kelley in [3] and on the Lp spaces of measure theory by Rota [5]. The motivation for this paper arose out of the latter paper. The aim of this paper is to prove a generalisation of Rota's main representation theorem (every average is a conditional expectation) in the context of a ‘non commutative integration’. This context is as follows. Let be a finite von Neumann algebra and ϕ a faithful normal finite trace on such that ϕ(I) = 1, where I is the identity of . We can construct the Banach spaces Lp (, ϕ), where 1 ≤ p < °, with norm ∥x∥p = ϕ(÷x÷p)1/p, of possibly unbounded operators affiliated with , as in [9]. We note that is dense in Lp(, ϕ). These spaces share many of the features of the Lp spaces of measure theory; indeed if is abelian then Lp(,ϕ) is isometrically isomorphic to Lp of some measure space.

We characterize those non-negative, measurable functions ψ on [0, 1] and positive, continuous functions ω1 and ω2 on ℝ+ for which the generalized Hardy–Cesàro operator

$$\begin{equation*}(U_{\psi}f)(x)=\int_0^1 f(tx)\psi(t)\,dt\end{equation*}$$

We prove a natural factorization of supersolvable groups and then we give another characterization of them in connection with the Fitting subgroup. Applying these theorems we describe the structure of some subclasses of supersolvable groups.

Voevodsky has conjectured that numerical equivalence and smash-equivalence coincide for algebraic cycles on any smooth projective variety. Building on work of Vial and Kahn–Sebastian, we give some new examples of varieties where Voevodsky's conjecture is verified.

Throughout this note, rings will mean associative rings with identity and all modules are unital. A ring R is called right artinian if R satisfies the descending chain condition for right ideals. It is known that not every ideal of a right artinian ring is right artinian as a ring, in general. However, if every ideal of a right artinian ring R is right artinian then R is called hereditarily artinian. The structure of hereditarily artinian rings was described completely by Kertész and Widiger [5] from which, in the case of rings with identity, we get:

A ring R is hereditarily artinian if and only if R is a direct sum S ⊕ F of a semiprime right artinian ring S and a finite ring F.

The Banach space lp(p≥1) is the space of all infinite sequences x = (x1, x2, x3, …) of real or complex numbers such that is convergent, with the norm defined by

The unit sphere S of lp is the set of all points x ∈ lp with ∥x∥ ≤ 1 and the sphere of radius a ≥ 0 centred at y ∈ lp is denoted by Sa(y), so that

In this note we consider the question as to which central division algebras occur as the Clifford algebra of a quadratic form over a field. Non-commutative ones other than quaternion division algebras can occur and it is also the case that there are certain central division algebras D which, while not themselves occurring as a Clifford algebra, are such that some matrix ring over D does occur as a Clifford algebra. We also consider the further question as to which involutions on the division algebra can occur as one of two natural involutions on the Clifford algebra.

Let

be an indefinite quadratic form with real coefficients. A well-known result, due to Birch, Davenport and Ridout [1], [5] and [6], states that if n ≥21 then for any ε > 0 there is an integer vector x ≠O such that

Recently [3] we have quantified this result, obtaining a function g(n) such that g(n)→ ½ as n n→ ∞ and such that for any η > 0 and all large enough X there is an integer vector x satisfying

where |x| = max |xi|and the implicit constant in Vinogradov's ≪-notation is independent of X.

There are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category \[\mathcal{A}\], given an object X and a set of objects \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\], an isomorphism \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] can be built, where \[Ex{t^{\text{n}}}\] is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to \[Ex{t^{\text{n}}}\] that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits in AB4 abelian categories with not necessarily enough projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.

For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p. We give a corollary that settles Iizuka’s conjecture for the case n=1 and p>2.