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In this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most 2. We then give a complete characterization of the n-centre of that unit group. The n-centre of the unit group is either the centre or the second centre (for n ≥ 2).

Given function Ω on ℝn , we define the fractional maximal operator and the fractional integral operator by and respectively, where 0 < α < n. In this paper we study the weighted norm inequalities of MΩα and TΩα for appropriate α, s and A(p, q) weights in the case that Ω∈ Ls(Sn-1)(s> 1), homogeneous of degree zero.

The genus series for maps is the generating series for the number of rooted maps with a given number of vertices and faces of each degree, and a given number of edges. It captures topological information about surfaces, and appears in questions arising in statistical mechanics, topology, group rings, and certain aspects of free probability theory. An expression has been given previously for the genus series for maps in locally orientable surfaces in terms of zonal polynomials. The purpose of this paper is to derive an integral representation for the genus series. We then show how this can be used in conjunction with integration techniques to determine the genus series for monopoles in locally orientable surf aces. This complements the analogous result for monopoles in orientable surf aces previously obtained by Harer and Zagier. A conjecture, subsequently proved by Okounkov, is given for the evaluation of an expectation operator acting on the Jack symmetric function. It specialises to known results for Schur functions and zonal polynomials. 1. Introduction. Although the study of embeddings of graphs in surfaces is less well developed for locally orientable surfaces than it is for orientable surfaces, there are compelling algebraic and combinatorial reasons for studying them jointly. From the algebraic point of view it has been shown (5) that the genus series for maps in these two cases corresponds to the instances b =1 0o f

Let M(X, Y) denote the space of all continuous functions between X and Y and Mf (X, Y) the path component corresponding to a given map f : X Y.W hen X and Y are classical flag manifolds, we prove the components of M(X, Y) corresponding to "simple" maps f are classified up to rational homotopy type by the dimension of the kernel of f in degree two cohomology. In fact, these components are themselves all products of flag manifolds and odd spheres. 1. Introduction. When X and Y are flag manifolds or, more generally, F0-spaces (simply connected finite complexes with finite-dimensional rational homotopy and no rational cohomology in odd degrees), the rational classification problem for components of the function space M(X, Y) intersects two basic areas of research. First, W. Meier (10) proved the identity component M1(X, X )f or an F 0-space X is rationally a product of odd spheres if and only if the rational Serre spectral sequence collapses for any orientable fi- bration with fibreX. Thus identifying the rational homotopy type of this particular func- tion space component is equivalent to resolving the Halperin conjecture for F0-spaces. Second, the rational classification of components is directly related to the problem of describing the set (X , Y ) of maps between the rationalizations of X and Y. For conve- nience, we denote this set by (X, Y) .W hen X Yis a generalized complex flag mani- fold this latter problem has been studied extensively by several authors (see (4,9)) with particular emphasis on the group E(X ) of rational self-equivalences. By (1, Corollary 3.6) the set (X, Y) for F0-spaces is in bijection with Hom H (Y, ), H (X, ) and so determining its structure is a purely algebraic problem. Nonetheless, there appears to be no general structure theorem in the literature for the rational maps between two different flag manifolds. In this paper, we focus on the large class of "simple" and "signed-simple" maps between flag manifolds and classify the components corresponding to these maps in the complex and symplectic cases. Let X G1 T and Y G2 T be flag manifolds where G1 and G2 are compact, connected Lie groups and T denotes a maximal torus of appropriate rank. By (2), H (Gi T) Bi Ji, i 1, 2, where Bi is the polynomial algebra on rank(Gi) variables generated in degree two and Ji is the ideal consisting of polynomials invariant under the

Symplectic and orthogonal Schur functions can be defined combinatorially in a manner similar to the classical Schur functions. This paper demonstrates that they can also be expressed as determinants. These determinants are generated using planar decompositions of tableaux into strips and the equivalence of these determinants to symplectic or orthogonal Schur functions is established by Gessel-Viennot lattice path techniques. Results for rational (also called composite) Schur functions are also obtained.

In [1], Jean-Philippe Anker conjectures an upper bound for the heat kernel of a symmetric space of noncompact type. We show in this paper that his prediction is verified for the space of positive definite n × n real matrices.

Let G be a finite π-separable group, where π is a set of primes. The π-partial characters of G are the restrictions of the ordinary characters to the set of π-elements of G. Such an object is said to be irreducible if it is not the sum of two nonzero partial characters and the set of irreducible π- partial characters of G is denoted Iπ(G). (If p is a prime and π = p′, then Iπ(G) is exactly the set of irreducible Brauer characters at p.)From their definition, it is obvious that each partial character φ ∊ Iπ(G) can be “lifted” to an ordinary character χ ∊ Irr(G). (This means that φ is the restriction of χ to the π-elements of G.) In fact, there is a known set of canonical lifts Bπ(G) ⊆ Irr(G) for the irreducible π-partial characters. In this paper, it is proved that if 2 ∉ π, then there is an alternative set of canonical lifts (denoted Dπ(G)) that behaves better with respect to character induction.An application of this theory to M-groups is presented. If G is an M-group and S ⊆ G is a subnormal subgroup, consider a primitive character θ ⊆ Irr(S). It was known previously that if |G : S| is odd, then θ must be linear. It is proved here without restriction on the index of S that θ(1) is a power of 2.

We show that , the E-homology of the Ω-spectrum for P(n), is an E* free Hopf ring for E a complex oriented theory with In sent to 0. This covers the cases and . The generators of the Hopf ring are those necessary for the stable result. The motivation for this paper is to show that P(n) satisfies all of the conditions for the machinery of unstable cohomology operations set up in [BJW95]. This can then be used to produce splittings analogous to those for BP done in [Wil75]

An implicitly defined second order ODE is said to be singular if the second derivative cannot be smoothly written in terms of lower order variables. The standard existence and uniqueness theory cannot be applied to such ODE and the graphs of solutions may fail to be regular curves (i.e., the solutions may have isolated C0-points or may fail to be single valued). In this paper we describe a local analysis for a large class of implicit second order ODE whose singular points satisfy a regularity condition. Within this class of ODE there is a secondary notion of (contact) singularity which is analogous to rest points for regular ODE. Theorems 5, 6, 7 and 8 produce invariants for these singularities which control the existence, uniqueness and the level of regularity in solutions.

This paper shows that for the Moore spectrum MG associated with any abelian group G, and for any positive integer n, the order of the Postnikov k-invariant kn+1(MG) is equal to the exponent of the homotopy group πnMG. In the case of the sphere spectrum S, this implies that the exponents of the homotopy groups of S provide a universal estimate for the exponent of the kernel of the stable Hurewicz homomorphism hn: πnX → En(X) for the homology theory E*(—) corresponding to any connective ring spectrum E such that π0E is torsion-free and for any bounded below spectrum X. Moreover, an upper bound for the exponent of the cokernel of the generalized Hurewicz homomorphism hn: En(X) → Hn(X; π0E), induced by the 0-th Postnikov section of E, is obtained for any connective spectrum E. An application of these results enables us to approximate in a universal way both kernel and cokernel of the unstable Hurewicz homomorphism between the algebraic K-theory of any ring and the ordinary integral homology of its linear group.

An additive subgroup P of a skew field F is called a prime of F if P does not contain the identity, but if the product xy of two elements x and y in F is contained in P, then x or y is in P. A prime segment of F is given by two neighbouring primes P1 ⊃ P2; such a segment is invariant, simple, or exceptional depending on whether A(P1) = {a ∈ P1 | P1aP1 ⊂ P1} equals P1, P2 or lies properly between P1 and P2. The set T(F) of all primes of F together with the containment relation is a tree if |T(F)| is finite, and 1 < |T(F)| < ∞ is possible if F is not commutative. In this paper we construct skew fields with prescribed types of sequences of prime segments as skew fields F of fractions of group rings of certain right ordered groups. In particular, groups G of affine transformations on ordered vector spaces V are considered, and the relationship between properties of Dedekind cuts of V, certain right orders on G, and chains of prime segments of F is investigated. A general result in Section 4 describing the possible orders on vector spaces over ordered fields may be of independent interest.

In his first and lost notebooks, Ramanujan recorded several values for the Rogers-Ramanujan continued fraction. Some of these results have been proved by K. G. Ramanathan, using mostly ideas with which Ramanujan was unfamiliar. In this paper, eight of Ramanujan's values are established; four are proved for the first time, while the remaining four had been previously proved by Ramanathan by entirely different methods. Our proofs employ some of Ramanujan's beautiful eta-function identities, which have not been heretofore used for evaluating continued fractions.

The well-known Tschirnhausen transformation, , eliminates the second term of the polynomial xn + axn-l + …. By a mere repeated application of this transformation, one can decide whether a given element of k[[x,y]] is prime (irreducible) or not. Here K is an algebraically closed field of characteristic 0. A generalised version of Hensel's Lemma is developed for the proofs. The entire paper can be understood by undergraduate students.

In this paper, we study two generalizations of the Stirling numbers of the first and second kinds, inspired from their combinatorial interpretation in terms of 0-1 tableaux. They are the 𝔄-Stirling numbers and the partial Stirling numbers. In particular, we give a q and a p, q-analogue of convolution formulae for Stirling numbers of the second kind, due to Chen and Verde-Star, and we extend these formulae to Stirling numbers of the first kind. Included in this study are the a, d-progressive Stirlingnumbers, corresponding to 0-1 tableaux with column lengths from an arithmetic progression ﹛a + id﹜i≥0.

Let p be a prime number, and let F be a free pro-p group of rank two. Consider an automorphism α of F of finite order m, and let FixF(α) = {x ∈ F | α(x) = x} be the subgroup of F consisting of the elements fixed by α. It is known that if m is prime to p and α = idF, then the rank of FixF(α) is infinite. In this paper we show that if m is a finite power pr of p, the rank of FixF(α) is at most 2. We conjecture that if the rank of F is n and the order of a is a power of α, then rank (FixF(α)) ≤ n.

The intersection IW(G) of the normalizers of the infinite subnormal subgroups of a group G is a characteristic subgroup containing the Wielandt subgroup W(G) which we call the generalized Wielandt subgroup. In this paper we show that if G is infinite, then the structure of IW(G)/ W(G) is quite restricted, being controlled by a certain characteristic subgroup S(G). If S(G) is finite, then so is IW(G)/ W(G), whereas if S(G) is an infinite Prüfer-by-finite group, then IW(G)/W(G) is metabelian. In all other cases, IW(G) = W(G).

In this paper, we prove that any sequence of polynomials (pn)n for which dgr(pn) = n which satisfies a (2N + l)-term recurrence relation is orthogonal with respect to a positive definite N × N matrix of measures. We use that result to prove asymptotic properties of the kernel polynomials associated to a positive measure or a positive definite matrix of measures. Finally, some examples are given.

In the paper we find, for certain values of the parameters, the spaces of multipliers (H(p, q, α), H(s, t, β) and (H(p, q, α), ls), where H(p, q, α) denotes the space of analytic functions on the unit disc such that . As corollaries we recover some new results about multipliers on Bergman spaces and Hardy spaces.

A module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:(1) R is a left perfect and every weakly projective right R-module is weakly injective.(2) R is a direct sum of matrix rings over local QF-rings.(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.(4) R is right artinian ring and every weakly injective right R-module is weakly projective.(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

Compact C*-convex subsets of Mn correspond exactly to n-th matrix ranges of operators. The main result of this paper is to discover the “right” analog of linear extreme points, called structural elements, and then to prove a generalised Krein-Milman theorem for C*-convex subsets of Mn. The relationship between structural elements and an earlier attempted generalisation, called C*-extreme points, is examined, solving affirmatively a conjecture of Loebl and Paulsen [8]. An improved bound for a C* -convex version of the Caratheodory theorem for convex sets is also given.

For any variety of groups, the relative inner rank of a given groupG is defined to be the maximal rank of the -free homomorphic images of G. In this paper we explore metabelian inner ranks of certain one-relator groups. Using the well-known Quillen-Suslin Theorem, in conjunction with an elegant result of Artamonov, we prove that if r is any "Δ-modular" element of the free metabelian group Mn of rank n > 2 then the metabelian inner rank of the quotient group Mn/(r) is at most [n/2]. As a corollary we deduce that the metabelian inner rank of the (orientable) surface group of genus k is precisely k. This extends the corresponding result of Zieschang about the absolute inner ranks of these surface groups. In continuation of some further applications of the Quillen-Suslin Theorem we give necessary and sufficient conditions for a system g = (g1,..., gk) of k elements of a free metabelian group Mn, k ≤ n, to be a part of a basis of Mn. This extends results of Bachmuth and Timoshenko who considered the cases k = n and k < n — 3 respectively.

In this paper the Corwin's conjecture is proved, which says that if d is a function analytic near ∞, then the hypoellipticity of the convolution operator Ad, defined by for every u ∊ S'(ℝn), implies that P(x)/ logx → ∞ as x → ∞, where P(x) is the distance from x ∊ ℝn to the set of complex zeros of d.

It is well known that the length Zn of the longest head run observed in n tosses with a fair coin is approximately equal to log2n with a stochastically bounded remainder term. Though — log2n does not converge in law, in the present paper it is shown to have almost sure limit distribution in the sense of the a. s. central limit theorem having been studied recently. The results are formulated and proved in a general setup covering other interesting problems connected with patterns and runs such as the longest monotone block or the longest tube of a random walk.

In the present paper we completely describe the isomorphisms between projective elementary groups PSLn and PSLm (n ≥ 2, m ≥ 2) over division rings. It was found that such groups can be isomorphic only if n = m; the division rings are isomorphic or anti-isomorphic, except for the following groups:PSL(2,F7) and PSL(3,F2); PSL(2, F4) and PSL(2,F5).The idea is based on a deepening of the classical Hua's approach. This problem has been solved independently by H. Ren, Z. Wan and X. Wu using a different way

Let be independent Ornstein-Uhlenbeck processes and In this paper the law of iterated logarithm for X(t, n)is considered. The results obtained improve those of Csorgő and Lin(1988) and Schmuland(1987).

In this paper we prove an inequality that involves the minimal crossing number and the braid index of links by estimating Murasugi and Przytycki’s index for a planar bipartite graph.

For graphs A, B and a positive integer r, the relation means that whenever Δ is an r-colouring of the vertices of A, then there is an embedding ϕ of B into A such that Δ ∘ ϕ is constant. A class of graphs has the Ramsey property if, for every , there is an such that . For a given finite graph G, let Forb(G) denote the class of all finite graphs which do not embed G. It is known that, if G is 2-connected, then Forb() has the Ramsey property, and Forb(G) has the Ramsey property if and only if Forb(G) also has the Ramsey property. In this paper we show that if neither G nor its complement is 2-connected, then either (i) G has a cut point adjacent to every other vertex, or (ii) G has a cut point adjacent to every other vertex except one. We show that Forb(G) has the Ramsey property if G is a path of length 2 or 3, but that Forb(G) does not have the Ramsey property if (i) holds and G is not the path of length 2.

Let h: [0,1] → R be such that and define .In 1966, Erdős [8] proved that holds for almost all n, which by using a simple argument implies that in the case h(u) = u, for almost all n, He further obtained that, for every z > 0 and almost all n, and that where ϕ, ψ, are continuous distribution functions. Several other results concerning the normal growth of prime factors of integers were obtained by Galambos [10], [11] and by De Koninck and Galambos [6].Let χ = ﹛xm : w ∈ N﹜ be a sequence of real numbers such that limm→∞ xm = +∞. For each x ∈ χ let be a set of primes p ≤x. Denote by p(n) the smallest prime factor of n. In this paper, we investigate the number of prime divisors p of n, belonging to for which Th(n,p) > z. Given Δ < 1, we study the behaviour of the function We also investigate the two functions , where, in each case, h belongs to a large class of functions.

In this paper we consider the relationship between real quadratic fields, their class numbers and the continued fraction expansion of related ideals, as well as the prime-producing capacity of certain canonical quadratic polynomials. This continues and extends work in [10]–[31] and is related to work in [3]–[4].

This paper is devoted to settling the following problem on (infinite, partially) ordered sets: Is there always a partition (2-coloring) of an ordered set X so that all nontrivial maximal chains of X meet both classes (receive both colors)? We show this is true for all countable ordered sets and provide counterexamples of cardinality N3. Variants of the problem are also considered and open problems specified.

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

One of the important invariants of a р-block B of a group algebra is ℓ (B), the number of non-isomorphic simple B-modules. A number of authors calculated ℓ (B) for various types of defect groups of B. In particular, by Olsson [6], it has been proved that if p = 2 and the defect groups of the block B are dihedral or semi-dihedral or generalized quaternion, then ℓ (B) is at most 3. In this paper, we restrict our attention to the principal p-block B0 of a finite р-solvable group with ℓ (B0) ≤ 3. Let Γ be a finite р-solvable group and k a splitting field for Γ with characteristic р.

In this paper we show that every locally finite strongly Abelian variety satisfies the Hamiltonian property. An algebra is Hamiltonian if every one of its subuniverses is a block of some congruence of the algebra. A counterexample is provided to show that not all strongly Abelian varieties are Hamiltonian.

In the paper [KIS2], C. Kiselman studied the boundary smoothness of the vector sum of two smoothly bounded convex sets A and B in . He discovered the startling fact that even when A and B have real analytic boundary the set A + B need not have boundary smoothness exceeding C20/3 (this result is sharp). When A and B have C∞ boundaries, then the smoothness of the sum set breaks down at the level C5 (see [KIS2] for the various pathologies that arise).

This paper is a continuation of [3] which initiated a systematic study of sufficient conditions for the weighted interpolation inequality of sum form, 1.1 to hold. Here ϕ, θ are non-negative functions of m, j, p, q, r, Ω is a bounded or unbounded domain in Rn, ∊ belongs to an interval Γ=(0, ∊0), u is in a certain Banach space E(Ω), and N, W, P are measurable real functions satisfying N≧ 0, W, P > 0, as well as additional conditions stated below. Finally the constant K does not depend on u although it may depend on the other parameters.

Throughout this paper, all rings have the identity 1 and ring homomorphisms are assumed to preserve 1. We use p to denote a prime integer and F to denote a field of characteristic p. For an element α in F, we setA = F[ϰ]/(ϰp - α)F[ϰ].Moreover, by D and R, we denote the derivation of A induced by the ordinary derivation of F[ϰ] and the skew polynomial ring A[X,D] where aX = Xa+D(a) (a ∈ A), respectively (cf. [2]).In [3], R. W. Gilmer determined all the B-automorphisms of B[X] for any commutative ring B. Since then, some extensions or generalizations of his results have been obtained ([1], [2] and [5]). As to the characterization of automorphisms of skew polynomial rings, M. Rimmer [5] established a thorough result in case of automorphism type, while M. Ferrero and K. Kishimoto [2], among others, have made some progress in case of derivation type.

Recently, M. Hofmeister [4] counted all nonisomorphic double coverings of a graph by using its Ζ2 cohomology groups, and J. Kwak and J. Lee [5] did the same work for some finite-fold coverings. In this paper, we give an algebraic characterization of isomorphic graph bundles, from which we get a formula to count all nonisomorphic graph-bundles. Some applications to wheels are also discussed.

In this paper, we are primarily concerned with the behaviour of the centre with respect to the completion process for von Neumann regular rings at the pseudo-metric topology induced by a pseudo-rank function.Let R be a (von Neumann) regular ring, and N a pseudo-rank function (all terms left undefined here may be found in [6]). Then N induces a pseudo-metric topology on R, and the completion of R at this pseudo-metric, , is a right and left self-injective regular ring. Let Z( ) denote the centre of whatever ring is in the brackets. We are interested in the map .If R is simple, Z(R) is a field, so is discrete in the topology; yet Goodearl has constructed an example with Z(R) = R and Z(R) = C [5, 2.10]. There is thus no hope of a general density result.

The object of this paper is to determine the general solution of the functional equationFEwhere α is multiplicative. It turns out that non-trivial embeddings of the reals in the complex generate some interesting solutions.In many applications, various special cases of (FE) have occurred ([1,3, 4, 6, 10, 11, 14]). The special case where f = g = h = k and α = the identity map is known as the fundamental equation of information, and has been extensively investigated by many authors ([5]). The case where f = g = h = k and α is multiplicative was treated in [13, 14]. The general solution of (FE) when α(1 – x) = (1 – x)β has been obtained in [9], except when β = 2.

The graphs considered in this paper are finite and have no loops or multiple edges. If G is such a graph, we denote its vertex set by VG and its edge set by EG. If X and Y are disjoint subsets of VG, we define δ (X, Y) to be the set of edges of G that join a vertex in X to one in Y.

Let X be an infinite topological space, let 𝔫 be an infinite cardinal number with 𝔫 ≦ |X|. The basic problem in this paper is to find the number of closed sets in X of cardinality 𝔫. A complete answer to this question for the class of metrizable spaces has been given by A. H. Stone in [31], where he proves the following result. Let X be an infinite metrizable space of weight 𝔪, let 𝔫 ≦ |X|.

All spaces considered in this paper will be metric spaces. A subset A of a space X is called a fixed point set of X if there is a map (i.e., continuous function) ƒ: X → X such that ƒ(x) = x if and only if x ∈ A. In [22] L. E. Ward, Jr. defines a space X to have the complete invariance property (CIP) provided that each of the nonempty closed subsets of X is a fixed point set of X. The problem of determining fixed point sets of spaces has been investigated in [14] through [20] and [22]. Some spaces known to have CIP are n-cells[15], dendrites [20], convex subsets of Banach spaces [22], compact manifolds without boundary [16], and a class of polyhedra which includes all compact triangulable manifolds with or without boundary [18].

Section 1 of this paper contains all the work which deals exclusively with generalizations of Hadamard matrices. The non-existence theorem proven here (Theorem 1.10) generalizes a theorem of Hall and Paige [15] on the non-existence of complete mappings in certain groups.In Sections 2 and 3, we consider the duals of (Hanani) transversal designs; these dual structures, which we call (s, r, µ)-nets, are a natural generalization of the much studied (Bruck) nets which in turn are equivalent to sets of mutually orthogonal Latin squares. An (s, r, µ)-netis a set ofs2µpoints together withrparallel classes of blocks. Each class consists ofsblocks of equal cardinality. Two non-parallel blocks meet in preciselyµpoints. It has been proven thatris always less than or equal to (s2µ– l) / (s– 1).

This paper arises in the attempt to solve the following problem related to the Zariski Problem. Let A be a polynomial ring in three variables over a field, . Suppose there is a subring B of A such that k ⊆ B and there is variable t over B such that B[t] = A. Then is it true that B is a polynomial ring over k?

Let G be a subgroup of the general linear group GL(n, Q) over the rational field Q, and consider its action by right multiplication on the vector space Qn of n-tuples over Q. The present paper investigates the question of how we may constructively determine the orbits and stabilizers of this action for suitable classes of groups. We suppose that G is specified by a finite set {x1, …, xr) of generators, and investigate whether there exist algorithms to solve the two problems:(Orbit Problem) Given u, v ∊ Qn, does there exist x ∊ G such that ux = v; if so, find such an element x as a word in x1, …, xr and their inverses.(Stabilizer Problem) Given u, v ∊ Qn, describe all words in x1, …, xr and their inverses which lie in the stabilizer

The aim of this paper is to study a decomposition of finite T0 spaces into topological entities called chains and cells. These objects behave like complete units under homeomorphisms and they appear to be useful in investigating certain aspects of finite spaces. As an elementary illustration of how these entities can be used, the concept of an A2 space is introduced (in the next paragraph) and it is demonstrated that the order of the automorphism group of an A2 space is expressible as 2t, for some t ≧ 0.

The purpose of this paper is to continue the investigation into the relationships amongst Massey products, lower central series of free groups and the free differential calculus (see [4], [9], [12]). In particular we set forth the notion of a universal Massey product ≪α1, …, αk≫, where the αi are one dimensional cohomology classes. This product is defined with zero indeterminacy, natural and multilinear in its variables.In order to state the results we need some notation. Throughout F will denote the free group on fixed generators x1, …, xn andwill denote the lower central series of F. If I = (i1, …, ik) is a sequence such that 1 ≦ i1, …, ik ≦ n then ∂1 is the iterated Fox derivative and , where is the augmentation. By convention we set ∂1 = identity if I is empty.

In this paper we study the existence problem for topological actions of finite groups on euclidean spaces Rn+k which are free outside a fixed point set Rk (embedded as a vector subspace). We refer to such an action as a semi-free action on (Rn+k, Rn) and note that all our actions will be assumed orientation-preserving.Suppose the finite group π acts semi-freely on (Rn+k, Rn), then it acts freely on (Rn+k – Rn) = Sn–l × Rk+1. Since this space is homotopy equivalent to Sn–l, π will have periodic integral cohomology and n will be a multiple of the period. In fact the orbit spaceis a finitely-dominated Poincaré complex of formal dimension n – 1 with π1W = π and as considered by Swan [41].

Coinitial graphs were used in [2; 3 ; 4] as a combinatorial tool in the Reidemeister- Schreier process in order to prove subgroup theorems for Fuchsian groups. Whitehead had previously introduced such graphs but used topological methods for his proofs [8; 9]. Subsequently Rapaport [7] and Iliggins and Lyndon [1] gave algebraic proofs of the results in [9], and AIcCool [5; 6] has further developed these methods so that presentations of automorphism groups could be found.In this paper it is shown that Whitehead automorphisms can be described by a “cutting and pasting” operation on coinitial graphs. Section 1 defines and gives some combinatorial properties of these operations, based on [1].

Let X = {x1, … xn} be a free generating set of the free group Fn and let H be the subgroup of Aut Fn consisting of those automorphisms α such that α(xi) is conjugate to xi for each i = 1, 2 , …, n. We call H the Z-conjugating subgroup of Aut Fn. In [1] Humphries found a generating set for the isomorphic copy H1 of H consisting of Nielsen transformationswhere each is conjugate to ui (see remark 1 below). The purpose of this paper is to find a presentation of H (and hence of H1).Let i ≠ j be elements of {1, 2, …, n}. We denote by (xi; xj) the automorphism of Fn which sends xi to and fixes xk if k ≠ i. Let S be the set of all such automorphisms.