In this survey article we give an exposition of some work on the power structure of p-groups; especially work of the author's. The titles of the three sections are: Section 1. p-groups which have regular power structure; Section 2. Some weaker power structure properties; Section 3. p-central series of p-groups.

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements:

(1) Fr(V) discriminates V. (1*) The Fs(V) satisfy the same universal sentences for all s≥r. Moreover, we have introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent:

(2) Fr(V) strongly discriminates V. (2*) The Fs(V) satisfy the same universal formulas for all s ≥ r whenever elements of Fr(V) are substituted for the unquantified variables. On the surface (2) and (2*) appear to be stronger conditions than (1) and (1*). However, we have shown that for particular varieties (of groups) (2) and (2*) are no stronger than (1) and (1*).

We solve the question raised by Barbosa and do Carmo as to whether there exists a complete, noncompact stably immersed surface in R3 with nonzero constant mean curvature. We show that such a surface is necessarily minimal, that is, its mean curvature is zero.

Let us consider the abstract equation

(0.1) L (ɛ) u + F (ɛ,u) = 0,

where F (ɛ,u) = 0 (|u|2) for ɛ near zero. In this paper we define a multiplicity depending only on L (ɛ) allowing us to obtain the following result: “Odd multiplicity entails bifurcation and, if the multiplicity is even, it is possible to find F (ɛ,u) such that the only solution to (0.1) near the origin are the trivial ones”.

In this note, we establish some results concerning the total torsion and the total absolute torsion of certain non-closed stereographically projected analytic curves. The method of proof involves only elementary techniques of integration, a periodicity argument and Liouville's Theorem.

We present a unified treatment of the band circulant matrices which occur in the periodic spline and histospline projection theory with equispaced knots. Explicit bounds for the norm of these matrices are given.

In this note translation-invariant Dirichlet forms on a commutative hypergroup are studied. The main theorem gives a characterisation of an invariant Dirichlet form in terms of the negative definite function associated with it. As an illustration constructions of potentials arising from invariant Dirichlet forms are given. The examples of one- and two-dimensional Jacobi hypergroups yield specifications of invariant Dirichlet forms, particularly in the case of Gelfand pairs of compact type.

In this paper several common fixed point theorems for four continuous mappings in Menger and metric spaces are proved. These mappings are assumed to satisfy some generalizations of the contraction condition.

An asymptotic formula is derived for the number of partitions of a large positive integer n into r unequal positive integer parts and maximal summand k. The number of parts has a normal distribution about its maximum, the largest summand an extreme-value distribution. For unrestricted partitions the two distributions coincide and both are extreme-valued. The problem of joint distribution of unrestricted partitions with r parts and largest summand k remains unsolved.

Narayana Rao, Rodabaugh, Wilke and Zemmer constructed a new class of finite translation planes from exceptional near-fields described by Dickson and Zassenhaus. These planes referred to as C-planes are not coordinatized by the generalized André systems. In this paper we compute the translation complement of the C-plane corresponding to the C-system III–1. It is found that the translation complement is of order 6912 and it divides the set of ideal points into two orbits of lengths 2 and 48.

It is shown that if a bounded linear operator A commutes with a prespectral operator T of class Γ, then A commutes with the resolution of the identity of class Γ for T, say P (·), if and only if A*(Γ) ⊆ [PC*Γ. Here A* is the dual operator of A and [Pc]*Γ is the linear span of the set {U*ξ U ε P (·) c, ξ ε Γ} where P (·) c denotes the commutant of the range of P (·).

In this note, following Kaplanasky's study of semi-automorphisms of rings and Herstein's study of semi-homomorphisms of groups, we present a general study of semi-homomorphisms of rings.

Coefficient and integral mean bounds are obtained for functions subordinate to close-to-convex functions.

An adjoint-triangle theorem contemplates functors P: C → A and T: A → B where T and TP have left adjoints, and gives sufficient conditions for P also to have a left adjoint. We are concerned with the case where T is conservative - that is, isomorphism-reflecting; then P has a left adjoint under various combinations of completeness or cocompleteness conditions on C and A, with no explicit condition on P itself. We list systematically the strongest results we know of in this direction, augmenting those in the literature by some new ones.

Given a category A, we consider the (often large) set Ref A of its reflective (full, replete) subcategories, ordered by inclusion. It is known that, even when A is complete and cocomplete, wellpowered and cowellpowered, the intersection of two reflective subcategories need not be reflective. Supposing that A admits (i) small limits and (ii) arbitrary (even large) intersections of strong subobjects, we prove that an infimum ∧iCi in Ref A must necessarily be the intersection ∩iCi. Accordingly Ref A is not in general, even for good A, a complete lattice. We show, however, under the same conditions on A, that Ref A does admit small suprema ∨iCi, given by the closure in A of the union ∪iCi under the limits of type (i) and (ii) above.

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

If H and M are right R-modules, H is M-injective if every R-homonorphism N → H, N a right R-submodule of M, can be extended to an R-homonorphism from M to H.

H is strongly M-injective if H is injective for inclusions whose cokernels are isomorphic to factor modules of M.

For the case of abelian groups H and M, one settles the questions “when is H M-injective” and “when is H strongly M-injective”. The latter can be characterized in terms of the vanishing of Ext. Results for general module categories are also given.