We prove that factor groups of cartesian powers of finite non-abelian simple groups cannot be countably infinite. Thisis not our main result, but it had been our original aim. The proof is based on a similar fact concerning σ-complete Boolean algebras, and on a representation of certain subcartesian powers of a group in its group ring over a Boolean ring. This representation, to which we give the name “Boolean power”, will be our central theme, and we begin with it.

The ultraspherical polynomial (x) of degree n and order λ is defined by for n = 0, 1, 2, …. Of these polynomials, the most commonly used are the Chebyshev polynomials Tn(x) of the first kind, corresponding to λ = 0; the Legendre polynomials Pn(x) for which λ = ½; and the Chebyshev polynomials Un(x) of the second kind (λ = 1). In the first case the standardisation is different from that given in equation (1), since.

This note deals with some properties of algebraic number fields generated by numbers having all their conjugates on a circle. In particular, it is shown that an algebraic number field is a CM-field if and only if it is generated over the rationals by an element (not equal to ±1) whose conjugate all lie on the unit circle.

Subject classification (Amer. Math. Soc. (MOS) 1970): 12 A 15, 12 A 40, 14 K 22.

Let G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

We give a complete picture of the interaction between the Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras, imposing the condition that standard filtrations of projective modules are finite, and develop a tilting theory for such algebras. Under the assumption on existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both the Ringel and Koszul duals belong to the same class, and that these two dualities on this class commute.

We introduce the notion of a strong representation of a semigroup in the monoid of endomorphisms of any mathematical structure, and use this concept to provide a theoretical description of the automorphism group of any semigroup. As an application of our general theorems, we extend to semigroups a well-known result concerning automorphisms of groups, and we determine the automorphism groups of certain transformation semigroups and of the fundamental inverse semigroups.

We show that generalized Gaussian estimates for selfadjoint semigroups (e-tA)t ∈ R+ on L2 imply Lp boundedness of Riesz means and other regularizations of the Schrödinger group (eitA)t ∈ R. This generalizes results restricted to semigroups with a heat kernel, which are due to Sjöstrand, Alexopoulos and more recently Carron, Coulhon and Ouhabaz. This generalization is crucial for elliptic operators A that are of higher order or have singular lower order terms since, in general, their semigroups fail to have a heat kernel.

This paper continues the investigation of semigroup constructions motivated by applications in data mining. We give a complete description of the error-correcting capabilities of a large family of clusterers based on Rees matrix semigroups well known in semigroup theory. This result strengthens and complements previous formulas recently obtained in the literature. Examples show that our theorems do not generalize to other classes of semigroups.

This paper reports on a facility of the ANU NQ program for computation of nilpotent groups that satisfy an Engel-n identity. The relevant details of the algorithm are presented together with results on Engel-n groups for moderate values of n.

Let (G) denote the lattice of all subgroups of a group G. By an -isomorphism (lattice isomorphism) of G onto a group H, we mean an isomorphism of (G) onto (H). By an -isomorphism (normaliser preserving -isomorphism) of G onto H, we mean an -isomorphism ø such that (Aø) = (A)ø for all A ∈ (G). In this paper, we study certain properties of groups which remain invariant under -isomorphisms.

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d and δ non-zero derivations of R, f (x1,…, xn) a multilinear polynomial over K.If

then f(x1,…,xnis central-valued on R.

In this work, we review some properties of twisted partial actions of Hopf algebras on unital algebras and give necessary and sufficient conditions for a twisted partial action to have a globalization. We also elaborate a series of examples.

The first index law, or addition theorem, is well known. The second is much less well known; but both have been found to be of importance in recent studies of hypergeometric integral equations. The first law has usually been considered only in the simple case of orders of integration which have positive real part, or in the context of generalized functions. Arising out of the need to manipulate expressions involving several fractional integrals and derivatives, our aim here is to establish both laws for all combinations of complex orders of integration and differentiation, and for nearly all functions for which the fractional derivatives involved exist as locally integrable functions.

In the present paper, we derive transcendence measures for the numbers log α, eβ, αβ, (log α1)/(log α2) from a previous lower bound of ours on linear forms in the logarithms of algebraic numbers.

Here we mean growth in the sense of Milnor and Gromov. After a brief survey of known results, we compute the growth series of the groups , with respect to generators {x, y}. This is done using minimal normal forms obtained by informal use of judiciously chosen rewrite rules. In both of these examples the growth series is a rational function, and we suspect that this is not the case for the Baumslag-Solitar group

.

The theme of this paper is a conjecture whose origins go back to work of B. H. Neumann ([5], Theorem 3·1) and his former students I. Wiegold and I. D. Macdonald ([7], [4]). B. H. Neumann proved that if there is a bound to the sizes of conjugacy classes in the group G. that is. if G is a BFC group. then the derived group G′ is finite; Wiegold. and later Macdonald. produced explicit upper bounds for G′ in terms of the maximum n of the sizes of the conjugacy classes in G. The coniecture is that where l(n) should be the arithmetic function1 λ(n) for a best possible bound, or l(n) may be interpreted as log2n for a smooth, monotonicorderof-magnitude estimate. However, the bounds produced in [7] and [4], and even the vastly better upper bounds proved bv Sheooerd and Wieeold [6] for soluble groups, are very much bigger than , and it is mv aim in this paper to take a first step towards closing the gap. This first, step is study of BFC p-groups: in a sequel I hope to show how the results proved here can be used to obtain improved bounds for arbitrary BFC groups.

Within five minutes of the start of my first meeting with Bernhard Neumann as his research student, late in 1954, he suggested the following problem to me. Let G be a group in which the cardinals of the classes of conjugate elements are boundedly finite with maximum η, say. Then the commutator subgroup G′ is finite [6], Is the order |G′| of G′ bounded in terms of n? I distinctly recall these words of Neumann: “That should provide us with a start, I think”. He was right: more than just a start, the problem has been a continuing stimulus to a study of questions in fields as far apart as permutation group theory ([7], [11], [12], [14], and some unpublished work of Peter M. Neumann) and multiplicator theory ([13], [2]), as well as attracting interest in its own right.

Consider a compact zero dimensional (profinite) monoid. While the group of units must be open, a regular D-class need not be open in the ideal it generates. This is the case if and only if the semigroup contains infinitely many copies of a certain semilattice composed of an increasing sequence of idempotents converging to an upper bound.

Using compactifications of free products, two generator compact monoids with these properties are constructed.

We consider hypersurfaces of En+1 whose position vector x satisfies Δx = Ax + B, where Δ is the induced Laplacian, and prove that these are open parts of minimal hypersurfaces, hyperspheres or generalized circular cylinders.

A geometric mass concerning supersingular abelian varieties with real multiplications is formulated and related to an arithmetic mass. We determine the exact geometric mass formula for superspecial abelian varieties of Hubert-Blumenthal type. As an application, we compute the number of the irreducible components of the supersingular locus of some Hubert-Blumenthal varieties in terms of special values of the zeta function.