Let G be a transitive permutation group on a set Ω of n points, and let P be a Sylow p-subgroup of G for some prime p dividing ∣G∣. If P has t long orbits and f fixed points in Ω, then it is shown that f ≦ tp − ip(n), where ip(n) = p – rp(n), rp(n) denoting the residue of n modulo p. In addition, groups for which f attains the upper bound are classified.

Let C be a nontrivial full subcategory of the category F of finite discrete groups and continuous homomorphisms, closed under subobjects, quotient and finite products. We consider the category PC of pro-C-groups and continuous homomorphisms (i.e. inverse limits of C-groups) which forms a variety in category PF of profinite groups and continuous homomorphisms. The study of pro-Cgroups is motivated by their occurrence as Galois groups of filed extensions in algebraic number thory (see Serre (1965)). The purpose of this paper is to study the tripleableness of the forgetful functors from PC to various underlying categories. It is also shown that PC is equivalent to the category of algebras of the theory of the forgetful functor from C to S (the category of sets and mappings).

Let R be a normal subgroup of the free group F, and set G = F/[R, R]. We assume that F/R is a torsion-free group which is either solvable and not cyclic, or has a non-trivial center and is not cyclic-by-periodic. Then any automorphism of G whose restriction to R/[R, R] is trivial is an inner automorphism, determined by some element of R/[R, R]. This result extends a theorem of Šmel'kin (1967).

Seidel (1959) established various boundary properties of holomorphic functions with spiral asymptotic paths. Especially, Theorem 4 which is the fundamental result of the paper; was generalized by Faust (1962), using essentially the same method that was employed by Seidel.

Starters with adders, in abelian groups of odd order, have been used widely in recent combinatorial constructions, most notably in the study of Room squares. In this paper, constructions of adders for the so-called “patterned starter” are described, for a large class of nonabelian groups of odd order.

Let K be a field and G a finite group with subgroup H. We say that (G H) is a K-free pair if whenever M is a finitely generated KG-module whose restriction MH is a free KH-module, then M is a free KG-module. In this paper pairs of groups with this property will be investigated.

Conditions are found for the existence of supplementary difference sets consisting of cosets of e-th power residues modulo a prime p. For e = 4, 6 and 8 all known pairs of sets are listed in the summary.

The well known theorem of E. H. Brown on the representation of Homotopy Functors is extended to a larger domain category. This is accomplished by requiring the functor to satisfy a further condition, that of being locally trivial: the condition is both necessary and sufficient for the extension and is satisfied by functors arising from locally trivial fibre bundles such as K-theory.

Let PA(n) denote the number of partitions of n into summands chosen from the set A = {a1, a2, …}. De Bruijn has shown that in Mahler's partition problem (aν = rν) there is a periodic component in the asymptotic behaviour of PA(n). We show by example that this may happen for sequences that satisfy aν ν and consider an analogous phenomena for partitions into primes. We then consider corresponding results for partitions into distinct summands. Finally we obtain some weaker results using elementary methods.

The pointwise order makes the group A(Ω) of order-preserving permutations of a totally-ordered set Ω a lattice-ordered group. We give some criteria for determining the compatible tight Riesz orders on A(Ω) in the case of Ω being a totally-ordered field, and then obtain various adjunctionshellip one between tight Riesz orders on A(Ω) and certain ideals of the fixed point lattice Φ(Ω), and a second between maximal tangents and certain filters of Φ(Ω). We also establish a correspondence between tight Riesz orders and first-order properties. Finally, we make use of our results to say what we can in the case of the automorphisms of the real field, and to pose several open problems.

The variety of quasigroups is universal for varieties of algebras of the most general kind in the sense that each such variety can be interpreted in a natural way in a suitably chosen subvariety of quasigroups. More precisely, for any algebra 〈A, f0, f1, f2, …〉 where f0, f1, f2, … is an arbitrary finite or infinite sequence of operations of finite rank, there exists a quasigroup 〈B,.〉 and polynormial operations F0, F1, F2, … over 〈B,·〉 such that 〈A, f0, f1, …〉 is a subalgebra of 〈B, F0, F1, …〉 satisfying exactly the same identities. Moreover, if there are only finitely many f0, f1, …, then 〈B1〉 can be taken so that its identities are recursive in those of 〈A, f0, f1, …〉, If 〈A, f0, f1, …〉 is a free algebra with an infinite number of free generators, then B can also be taken to coincide with A. This universal property of quasigroups has a number of consequences for their equational metatheory.

In (1957) Ward characterized trees as being certain compact idempotent commutative monotone mobs. The purpose of the present note is to obtain a similar characterization for the semitrees as studied by Muenzenberger and Smithson (1973, 1975).

New characterizations of maximal topologies are given for a class of topologies including the compact, Lindelöf and m-compact topologies.

Generalization to arbitrary join-extensions of a poset of a recent characterization of those which are complete lattices.

In Lee (submitted), the GW-integral (the generalized Riemann integral using Wiener measure) is defined. The object of this article is to define stochastic integral in the set up given in Lee (submitted). We also investigate the connection between the stochastic integral defined with the Legesgue counter part, the Paley-Wiener-Zygmund integral in Paley, Weiner and Zygmund (1933). Applications of the stochastic integral will be explained elsewhere.

We present in this paper a discussion on some stability properties of line graphs. After relating the semi-stability properties of the line graph of a graph to a concept of Sheehan, we proceed to deduce that, with fully characterised lists of exceptions, the line graphs of trees and unicyclic graphs are semi-stable. We then discuss the problem of deciding which line graphs are stable. Via a discovery of the finite number of graphs G such that both G and its complement have stable line graphs, we show that P4 is the only self-complementary graph whose line graph is stable.

Let ℒ be a normal base of a Tychonoff space X and ωℒ denote the Wallman-type (real-) compactification of X generated by ℒ. This Wallman-type compactification is known to associate with a unique proximity δ. A ℒ-filter ℒ is round if for each F ∈ ℒ there is an Fo ∈ ℒ there is an Fo ∈ ℒ such that Fo(X-F). A subset A of ω(£) is called a round subset of ω (£) iff for each Z ∈ ℒ, if C1w(x)Z contains A, then it is a neighborhood of A. Properties of round ℒ-filters and round sets of ω(ℒ) are introduced. We also prove that the intersection of all the free ℒ-ultrafilters is ℒ= {Z ∈ ℒ: C1x(X-Z) is compact} iff ω(ℒ) – X is a round subset of ω(ℒ) if ℒ is a separating nest generated intersection ring with property (α) then ω(ℒ) - v(ℒ) is a round subset of ω(ℒ).

Let M be a vector subspace of the Banach space C(Ω) of all real-valued continuous functions on a compact space Ω, and suppose that M contains a subset L and the constant functions. Then in order that L be dense in M it is necessary and sufficient that L, M satisfy a filtering property and that each m in M can be approximated on every two points of Ω by functions in L.

The Hardy classes for functions of bounded argument rotation and their derivatives are determined. In addition, if , then growth conditions for an are obtained.

Let G be a group, and associate with G a graph 1 γ = Γ (G) as follows: the vertices of Γ are the elements of G, and two vertices g, h of Γ are joined by an undirected edge if, and only if, g and h do not commute as elements of G, that is [g, h] ≠ 1 [where [g, h] is the commutator g−1h−1gh, and 1 is the unit element of the groups that occur as well as the integer, according to context].We are interested in complete subgraphs of Γ, or equivalently in sets of elements of G no two of which commute.