In this article, we obtain results on commutators in Sylow subgroups of the lower central series, extending the work of Dark and Newell [2], Rodney [12, 13] and Aschbacher and the author [1, 6, 7].

Some notation is required for the statement of the main results. Let r be a positive integer and define

and

where x1, …, xr, are elements in a group G. Let ΓrG = {[x1, …, xr]∣ x1 ∈ G} be the set of r-fold commutators in G. Then Lr,G = 〈ΓrG〉 is the rth term in the lower central series of G. Set L∞G = ∩ Lr,G.

We construct, for q a root of unity of odd order, an embedding of the projective special linear group PSL(n) into the group of bi-Galois objects over uq(sl(n))*, the coordinate algebra of the Frobenius–Lusztig kernel of SL(n), which is shown to be an isomorphism at n=2.

In this paper, we provide two new extensions to a lemma of Bernik (1983). Applications are also discussed.

A group G is called Dedekindian if every subgroup ofG is normal in G.

The structure of the finite Dedekindian groups is well-known [3, Satz 7.12]. They are either abelian or direct products of the form Q × A × B, where Q is the quaternion group of order 8, Ais abelian of odd order and exp(B) ≤ 2.

Let $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$.

1. (i) We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate

   $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$
   For each p, the constant e1/p is the best possible.

2. (ii) We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$,

   $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$
   and prove that the constant e is optimal.

We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its basic idempotents. We also give a few necessary and sufficient conditions for a semiperfect ring R, which cogenerates every 2-generated right R-module, to be right pseudo-Frobenius.

We construct cocompact lattices Γ'0 < Γ0 in the group G = PGLd$({\mathbb{F}_q(\!(t)\!)\!})$ which are type-preserving and act transitively on the set of vertices of each type in the building Δ associated to G. These lattices are commensurable with the lattices of Cartwright–Steger Isr. J. Math.103 (1998), 125–140. The stabiliser of each vertex in Γ'0 is a Singer cycle and the stabiliser of each vertex in Γ0 is isomorphic to the normaliser of a Singer cycle in PGLd(q). We show that the intersections of Γ'0 and Γ0 with PSLd$({\mathbb{F}_q(\!(t)\!)\!})$ are lattices in PSLd$({\mathbb{F}_q(\!(t)\!)\!})$, and identify the pairs (d, q) such that the entire lattice Γ'0 or Γ0 is contained in PSLd$({\mathbb{F}_q(\!(t)\!)\!})$. Finally we discuss minimality of covolumes of cocompact lattices in SL3$({\mathbb{F}_q(\!(t)\!)\!})$. Our proofs combine the construction of Cartwright–Steger Isr. J. Math.103 (1998), 125–140 with results about Singer cycles and their normalisers, and geometric arguments.

It is easy to imagine that a subvariety of a vector bundle, whose intersection with every fibre is a vector subspace of constant dimension, must necessarily be a sub-bundle. We give two examples to show that this is not true, and several situations in which the implication does hold. For example it is true if the base is normal and the field has characteristic zero. A convenient test is whether or not the intersections with the fibres are reduced as schemes.

Functorial semi-norms on singular homology give refined ‘size’ information on singular homology classes. A fundamental example is the ℓ1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the ℓ1-semi-norm.

Let Q be a complete local ring which has the same characteristic as its residue field P, and, for the present, let us denote by A the image of a subset A of Q under the natural homomorphism of Q onto P. Then a subfield F of Q is called a coefficient field if = P. It has been shown in [2] and in [3] that a complete equicharacteristic local ring, such as the above, always possesses at least one coefficient field; this is the embedding theorem for the equicharacteristic case.

If k denotes a number field and εm is the product of an elliptic curve ε with itself m times over k, then for each prime π where ε has non-degenerate reduction, the zeta factor ζ(επ'S) can be expressed as

Where |π| denotes the norm of π. It is a consequence of a conjecture of Tate [16] that if ε does not have complex multiplications, then the numbers are distributed according to the density function

that is, the density of the set of primes π such that – is

Following Craw, Maclagan, Thomas and Nakamura's works[2, 7] on Hilbert schemes for abelian groups, we give an explicit description of theHilbGℂ3 scheme for G = 〈diag(ϵ, ϵa, ϵr−a)〉 by a classification of all G-sets. We describe how the combinatorial properties of the fan of HilbGℂ3 relates to the Euclidean algorithm.

There are two well-known methods to build up algebras from given algebras, the direct and inverse limits, and a systematic account of these constructions may be found in [2]. It is known that every algebra can be represented as a direct limit of finitely generated algebras although in some cases the representation is trivial. Furthermore, Haimo [3] has established a certain inverse limit representation for the class of all infinite Boolean algebras which generalises, in actual fact, to the class of all infinite lattices with 1. The purpose of this note is to exhibit a certain nontrivial inverse limit representation which is peculiar to the class of infinite, complete Boolean algebras.

Let ℍ be the division ring of real quaternions. Let SL(2, ℍ) be the group of 2 × 2 quaternionic matrices $A={\scriptsize{(\begin{array}{l@{\quad}l} a & b \\ c & d \end{array})}}$ with quaternionic determinant det A = |ad − aca−1b| = 1. This group acts by the orientation-preserving isometries of the five-dimensional real hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, ℍ).

We shall extend results of Samuel [19] and Griffin [8, 9] about conditions which generalise the notion of valuation domain in a field. Let U be a commutative ring with identity, R a subring of U and L an R-submodule of U. The conditions we study have in common the property (EV), that the submodules L:x (x ∈ U) form a chain. We pay particular attention to the strongest of the conditions, viz, that L be a Manis valuation (MV) subring, i.e. having a prime ideal P such that (L, P) is a maximal pair in U (see [19], [16] and e.g. [4]). Such P is unique, being the union of all L:x such that x ∉ L, which we call P+(L) the centre of L. This set P+ plays a key role in the study of all our valuation conditions.

For any rotation-invariant positive regular Borel measure ν on the closed unit ball whose support contains the unit sphere , let L2a be the closure in L2 = L2(, dν) of all analytic polynomials. For a bounded Borel function f on , the Toeplitz operator Tf is defined by Tf(ϕ) = P(fϕ) for ϕ ∈ L2a, where P is the orthogonal projection from L2 onto L2a. We show that if f is continuous on , then Tf is compact if and only if f(z) = 0 for all z on the unit sphere. This is well known when L2a is replaced by the classical Bergman or Hardy space.

We show that every fibrewise map from a Serre microfibration to a Serre fibration is n-connected if it is fibrewise n-connected. This generalises a result of M. Weiss and related results by Bökstedt–Madsen and Galatius–Randal–Williams. We also discuss an application to configuration spaces.

In a recent paper [1] we showed that there is a (1,) -correspondence between the homomorphisms of an inverse semigroup S and its normal subsemigroups. The normal subsemigroup of S corresponding to and determining the homomorphism μ of S is the inverse image under μ of the set of idempotents of Sμ and is called the kernel of the homomorphism μ. The inverse image of each idempotent of Sμ is itself an inverse semigroup [1], and each such inverse semigroup is said to be a component of the normal subsemigroup determined by μ.

We use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)1 + o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over for some p≥m0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of .

In [1, Corollary 5], Figiel gives an elegant demonstration that the modulus ofconvexity δ in real Banach space X is nondecreasing, where

It is deduced from this that in fact δ(ɛ)/ɛ is nondecreasing [Proposition 3]. During the course of the proof [Lemma 4] it is stated that if v ∊ Sx is a local maximum on Sx of φ ∈Sx*, then v is a global maximum (φ(v) = 1). This is false; it could be that v is a global minimum. It is easy to construct such an example in R2 endowed with the maximum norm. What is true is that v is a global maximum of |φ|.