Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ , which is closely related to the cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{n}(x)$ and to primitive prime divisors of $q^{n}-1$ . Our definition of $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$ , we provide an algorithm for determining all pairs $(n,q)$ with $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)\leq cn^{k}$ . This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

Suppose that a ring is a sum of its nilpotent subrings. We use directed graphs to give new conditions sufficient for the whole ring to be nilpotent.

A universal power automorphism (Cooper [1]) of a group is an automorphism mapping every element x to a power xn for some fixed integer n. It is long known that a group admitting such an automorphism with n= −1, 2 or 3 must be Abelian. Miller [5] showed that for every other non-zero integral value of n there exist non-Abelian groups admitting a non-trivial universal power automorphism x→xn.

The Hopf bifurcation and homoclinic bifurcation of the quintic Hamiltonian system is analyzed under quintic perturbations by using unfolding theory in this paper. We show that a quintic system can have at least 29 limit cycles.

We first prove a basic theorem that if a set of polynomials satisfies an orthogonality relation with respect to integration, the set also satisfies an orthogonality relation with respect to summation. This result is then used to derive the Gaussian quadrature formula. The orthogonality relations give rise to interpolation formulas and a connection between the coefficients in these interpolation formulas is established. Finally, the analysis is used to get an estimate of the error in the Gaussian quadrature formula. Some error coefficients are evaluated in the cases where the orthogonal polynomials are those of Jacobi, Laguerre, Hermite and Bessel.

The most elementary problem of the calculus of variations consists in finding a single-valued function y(x), defined over an interval [a, b] and taking given values at the end points, such that the integral is stationary relative to all small weak variations of the function y(x) consistent with the boundary conditions. Since y′ occurs in the integrand, it is clear that I is only defined when y(x) is differentiable and accordingly when y(x) is continuous. Usually y′(x) is also continuous. Occasionally, however, the boundary conditions can only be satisfied and a stationary value of I found, by permitting y′ (x) to be discontinuous at a finite number of points. The arc y = y(x) will then possess ‘corners’ and the well-known Weierstrass-Erdmann corner conditions [1]must be satisfied at all such points by any function y (x) for which I is stationary. Arcs y = y (x) for which y′(x) is continuous except at a finite number of points, are referred to as admissible arcs. In this paper, we shall extend the range of admissible arcs to include those for which y(x) is discontinuous at a finite number of points.

It is shown that if G is a non-amenable group, then there are no non-zero translation invariant functionals on Lp(G) for 1 < p < ∞. Furthermore, if G contains a closed, non-abelian free subgroup, then there are no non-zero translation invariant functionals on C0(G). The latter is proved by showing that a certain non-invertible convolution operator on C0(G) is surjective.

In this paper a general concept of regularity for rings is defined. It is shown that every regularity determines in a natural way a subradical and a radical for rings. A wide class of regularities is constructed: the class of polynomial regularities. All well-known regularities, such as the Perlis-Jacobson regularity, the von Neumann regularity and many others, belong to this class. Special attention is paid to regularities which are elementary in the sense that the so-called unic and nullic polynomial regularities can be thought of as intersections of the elementary ones.

A three term recurrence relation is found for

when a + d = b + c. This includes the recurrence relations of Apéry associated with ζ(3), ζ(2) and log 2 as special or limiting cases.

A generalised factorial function (z: k)! is defined as an infinite product similar to the Euler product for z!, but with the sequences of integers replaced by the roots of F(z) = sin πz+kπz. It is proved that, apart from poles in (z) < 0, (z: k)! is analytic in both variables, and that F(z) may be expressed in the form F(z) = πz/(z: k)!(—z: k)!

This paper is an outgrowth of my old battle with the open sentence problem for the theory of finite groups. The unsolvability of the word problem for groups (cf. [1] and [4]) entails the undecidability of the open sentence problem for the elementary theory of groups and thus strengthens the original undecidability result for this theory (cf. [7]). The fact that the elementary theory of finite groups is also undecidable (cf. [2] and [6]) therefore justifies my interest in the open sentence problem for that theory. This paper contains a construction of groups that might lead to a negative solution.

We refer the reader to Hanna Neumann [7] for notation and other undefined terms. Let and denote the varieties of groups defined by the laws (xy)n=xnyn, [x, y]n=1 and [x, yn]respectively, where n is an integer. (n)-groups were termed “n-abelian” by R. Baer [1] and have been a subject matter of investigation by various authors (see [3], [5], [6] and the references therein). Recently KaluŽnin [5] has shown that , thus clarifying the relationship between U(n) and the familiar varieties. From the elementary inequalities(n ≠ 0,1)it is easily deduced that(see for instance [5]). If G = Cm Wr C∞, then cleary but for any Thus and . It is also easy to see that in general (see for instance [6] § 5.1) and we are led to ask

In this paper, we improve the Rosenbloom's fixed-point theorem and prove a related normality criterion. We also consider the corresponding unicity theorem for transcendental entire functions.

A ℂ-linear map θ (not necessarily bounded) between two Hilbert C*-modules is said to be ‘orthogonality preserving’ if 〈θ(x),θ(y)〉=0 whenever 〈x,y〉=0. We prove that if θ is an orthogonality preserving map from a full Hilbert C0(Ω)-module E into another Hilbert C0(Ω) -module F that satisfies a weaker notion of C0 (Ω) -linearity (called ‘localness’), then θ is bounded and there exists ϕ∈Cb (Ω)+ such that 〈θ(x),θ(y)〉=ϕ⋅〈x,y〉 for all x,y∈E.

Let X1, X2,…Xn, … be independent and identically distributed random variables, and write . In [2] Chung and Fuchs have established necessary and sufficient conditions for the random walk {Zn} to be recurrent, i.e. for Zn to return infinitely often to every neighbourhood of the origin. The object of this paper is to obtain similar results for the corresponding process in continuous time.

An S3-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S3-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G=PSL(2,q).

Kronecker classes of algebraci number fields were introduced by W. Jehne in an attempt to understand the extent to which the structure of an extension K: k of algebraic number fields was influenced by the decomposition of primes of k over K. He found an important link between Kronecker equivalent field extensions and a certain covering property of their Galois groups. This surveys recent contributions of Group Theory to the understanding of Kronecker equivalence of algebraic number fields. In particular some group theoretic conjectures related to the Kronecker class of an extension of bounded degree are explored.

N(G) denotes the near-ring of all continuous selfmaps of the topological group G (under composition and the pointwise induced operation) and N0(G) is the subnear-ring of N(G) consisting of all functions having the identity element of G fixed. It is known that if G is discrete then (a) N0(G) is simple and (b) N(G) is simple if and only if G is not of order 2. We begin a study of the ideal structure of these near-rings when G is a disconnected group.

A unified study is undertaken of finitely generated varieties HSP () of distributive lattices with unary operations, extending work of Cornish. The generating algebra () is assusmed to be of the form (P; ∧, ∨, 0, 1, {fμ}), where each fμ is an endomorphism or dual endomorphism of (P; ∧, ∨, 0, 1), and the Priestly dual of this lattice is an ordered semigroup N whose elements act by left multiplication to give the maps dual to the operations fμ. Duality theory is fully developed within this framework, into which fit many varieties arising in algebraic logic. Conditions on N are given for the natural and Priestley dualities for HSP () to be essentially the same, so that, inter alia, coproducts in HSP () are enriched D-coproducts.