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Let G toe a 2-transitive permutation group of a set !! of npoints and let P be a Sylow p-subgroup of G where p is aprime dividing \G\ . If we restrict the lengths of the orbitsof P , can we correspondingly restrict the order of P ? In theprevious two papers of this series we were concerned with thecase in which all P-orbits have length at most p ; in thesecond paper we looked at Sylow p-subgroups of a two pointstabiliser. We showed that either P had order p , orG > A , G = PSL(2, 5) with p = 2 , or G = M of degree 12with p = 3 . In this paper we assume that P has a subgroup Qof index p and all orbits of Q have length at most p . We2conclude that either P has order at most p , or the groupsare known; namely PSL(3, p) S G5 PGL(3, p) ,ASL(2, p) £ G 5 AGL(2, p) , G = PrL(2, 8) with p = 3 ,G = M with p = 3 , G = PGL(2, 5) with p = 2 , or G > A2with 3p - n < 2p ; all in their natural representations.Let G be a doubly transitive permutation group on a set ft of npoints and let P be a Sylow p-subgroup of G where p is a primedividing \G\ . The previous two papers [9, JO] were concerned with thesituation in which P has no orbit of length greater than p . We showedessentially that either G contains the alternating group or P has orderp . The general problem is the following:Received 21 May 1975.

In a recent note [6] the first author has announced the discovery of a family of transformation groups (G,), > o which have the following property: G, acts on the n! elements of the symmetric group ~ , and the number of its orbits is equal to the n-th tangent or secant number, according as n is odd or even. The purpose of this paper is to give a complete description of these groups. Applications to enumeration problems will appear in a subsequent paper. The tangent (or Euler) number are defined by the series expansion of tan u

An epimorphic extension of a ring R is a ring S for which there is a homomorphism 11: R ~ S which is both a monomorphism and an epimorphism in the category of rings. This paper is concerned with epimorphic extensions which are left-fiat in the sense that ~ induces on S the structure of a fiat left R-module; an example of such an extension is the (classical) ring of right quotients of R. Flat epimorphic extensions of commutative rings have been studied by Lazard [-8]; the more general situation in which ~ need not be monomorphic has been considered by Silver [-10]. Characterisations of left-fiat epimorphic extensions of a ring R are given in Section 3, from which it follows that such extensions are rings of right quotients of R in the sense of Utumi [121. Certain properties analogous to those possessed by classical rings of right quotients are also discussed. The principal result of the paper is Theorem 4.1, in which the existence of a left-fiat epimorphic extension P(R) of a ring R in which every other such extension can be uniquely embedded is established. It is shown, in Section 5, that, for a given ring R, P(R) is a semisimple artinian ring if and only if R contains no infinite direct sum of non-zero right ideals and the right singular ideal of R is zero. Finally, commutative semiprime rings are considered. It is shown that, if such a ring R contains all the idempotents of its complete ring Q (R) of quotients, then P(R) is the minimal regular subring of Q(R) which contains R. Examples of such rings are commutative semiprime rings which are integrally closed in Q (R) and commutative Baer rings. It has been drawn to my attention that some of the results of this paper are in the Notes [-3, 41 and [-91. Theorem 4.1 is announced, without proof, in [-91. A stronger result than Corollary 3.4 appears in [31 and [91 and Theorem 3.6 (iii) appears in [41. 2. Notation and Known Results

In this paper we develop a structure theory for certain projective modules which we denote as semi-perfect modules. By this notation we imply that these can be considered as a generalization of semi-perfect rings which are studied in a paper by BAss EIJ. A (right) semi-perfect ring R is defined by the property tha t every cyclic right-module over this ring or, equivalently, that every factor module of R modulo a right ideal, has a projective cover. Then, as a main result, BASS has proved tha t a ring R is semi-perfect if and only if R modulo its radical J(R) is semi-simple and decompositions of R/J(R) can be raised to R. This implies that the definition of a semi-perfect ring is independent of the side. We define a semi-perfect module M =.TV/R over an arbi trary ring R by the following two properties

Let G be a topological group and H a closed normal subgroup of G. !f G contains a closed subgroup K isomorphic to the quotient group G/H such that Hc~K={1}, the identity of G and that the map (h, k ) ~ h k is a homeomorphism of H x K onto G, then G is said to be a semidirect product of H and K, and we say that G splits over H. In recent years, a great deal has been learned about the structure of connected locally compact groups on the one hand and of compact totally disconnected groups on the other. The natural synthesis of these two seems to be locally compact groups which are compact modulo component of the identity. Let [C] denote the class of all such groups. In this paper, we are primarily concerned with study of [C]-groups. In studying the structure of [C]-groups, the chief purpose we have in mind is to see how far these groups are away from being semidirect products of connected locally compact groups and totally disconnected compact groups. Even though one can hope for such a splitting in rare instances, we are able to find a totally disconnected compact 'supplement ' to the identity component of a [C]-group. The proof of existence of such a supplement given in this paper uses the method which HOFMANN and MOSTERT [6] have developed in their investigation of splittings of topological groups over vector subgroups. As is well known, a study of [C]-groups sometimes requires aconsiderable knowledge on Lie algebras, since these groups may be approximated by Lie groups. Thus we devote the first chapter to an investigation of automorphisms of Lie algebras. Although this chapter is not an integral part of our topological investigations, it is closely related to the following chapters. The following are outlines of results in the order in which the chapters containing them occur.

Let 7~: (X, X ) -~ (M, (9) be a holomorphic mapping of an analytic space (X, JC) onto a complex manifold (M, (9). For each t eM, the fibre X t : r c l ( t ) is an analytic set in X. Let m t c (gt be the ideal sheaf of germs of all holomorphic functions which vanish on the point t and (mt o re) the idealsubsheaf in generated by g o 7c, g ~ mr. Let us put ~ , . . = ~ / ( m t o ~z) l X . Then (X t , 2/f t) is an analytic space. Thus a holomorphic mapping lr: X --~ M gives a family of analytic spaces X~, t ~ M . A family is reduced if all Xt are reduced analytic spaces (or the analytic spaces which are defined by Cartan and Serre). A family ~: X ~ M is flat at x ~ X if TOrl~(2/f~, (9]mt)=O where l r ( x ) t . Grauert and Kerner give in [43 an equivalent condition to flatness. In this paper, we shall give an another equivalent condition to flatness, that is. rc is flat at x if and only if the sequence of homomorphisms t~, ..., ~,, of ~'~x into 24~x is a regular sequence where (t I . . . . , t,) are local coordinates in a neighborhood of ~z (x). We shall give that if zr is an open mapping, then rc is flat. It is showed in [6] that a reduced family zr: X ~ M is an open mapping if and only if ~ is flat. Let re: X --+ M be a flat reduced family and let Xo be the fibre at a point 0e M. Then there exists a natural homomorphism of the sheaf ~-* (M) into the sheaf ~Y~(Xo), where J-* (M) is the sheaf which the hotomorphic vector field on M is lifted to X and 3-;~(Xo) is the sheaf of the infinitesimal deformation of Xo. If the natural homomorphism ~ * ( M ) ~ ( X 0) is surjective. ~ is called complete at x s X . Kerner proved in [6] that ~z is complete at a regular point of X. In this paper, we shall give that taking X ' = ~z-~ ( M 3 instead of X o in the theorem proved by Kerner, a similar consequence holds, where M' is any submanifold of M.

An important characteristic of an affine space is that each of its hyperplanes is itself an affine space of one dimension lower. This property plays an essential part in the analysis of the structure of affine spaces. The purpose of this paper is to develop an analogous structure theory for general affine designs. This enables the introduction of a concept of dimension for affine designs and also yields techniques for constructing affine designs from smaller ones. The initial sections introduce the required basic results and terminology. In w is proved the main decomposition theorem which demonstrates how an affine design may be constructed with a given decomposition into sets of smaller affine designs. The results of this paper form part of my doctoral thesis at the University of London. To my supervisor Professor D.R. Hughes, I am indebted for invaluable assistance and guidance.

In each term of the Adams spectral sequence [1, 2] Massey products [10] can be formed and in the E2-term it has been found convenient to describe specific elements by means of these products. In Theorem (1.1) of this paper a description is given of the action of the differential d r on the Massey product in E r. In homotopy an analogous construction to the Massey product is the Toda bracket [15] and Theorem (1.2) discusses the convergence of Massey products in the Adams spectral sequence to Toda brackets in homotopy. It states that, under favorable conditions, a Massey product that is formed from permanent cycles will itself contain a permanent cycle that is realized in homotopy by an element of the appropriate Toda bracket. The statement of both Theorems (1.1) and (1.2) will be found in w 1 and the proofs are given in w167 5, 6. The intervening sections contain various geometric and algebraic preliminaries. A previous account of Theorem (1.2) was given in [12] and I am grateful to Professor J.F.Adams for suggesting the problem to which this theorem provides a partial solution. In 1966, Ivanovskil announced, at the Moscow congress, various results concerning the Adams spectral sequence and there appears to be some overlap between these results and Theorem (1.2). More recently, in Chicago, Lawrence has developed an account of the higher Massey product behaviour in the Adams spectral sequence and his results generalize the main theorems of this paper.

In a recent paper, Petryshyn and Tucker [10] introduced the concept of a P~-compact ("generalized P-compact ') operator, acting on some subset of a Banach space with projectionally complete system, and they proved general existence theorems for fixed points of such operators. The purpose of our paper is to establish conditions which ensure that the set of fixed points of a P~-compact mapping is a continuum, i.e. a compact and connected set. If it cannot be assured that a given operator has a unique fixed point, the general idea consists in approximating this operator by more "regular" ones, in a sense which will be made precise below. In case of completely continuous or proper mappings on a general Banach space this question had been investigated e.g. by Aronszajn [1], Stampacchia [11], Krasnosel'skii and Sobolevskii [7], Browder and Gupta [2] and Vidossich [12]. Some applications to Volterra integraland hyperbolic partial differential equations are given in [5] and [6], while [7] includes some results for parabolic equations. In w 1 we introduce some basic definitions to be used in this paper. w 2 includes a relation between P~-compact and demicompact mappings that can be used in particular to simplify the proofs of some propositions in [10]. In w 3 we prove our first main result which is concerned with the set of fixed points of a P~-compact operator on a bounded closed convex set, while in the last paragraph this theorem is extended to an arbitrary bounded region by means of the degree of an A-proper mapping recently developped by Browder and Petryshyn [3]. In the main, our notation is like that in [10]. The author is indebted to Prof. W.V. Petryshyn who called his attention to A-proper mappings and to Prof. G. Stampacchia for having made a preliminary draft of [12] available to him.

The problem has been solved by Stoll for the case p ( n 1 ) [6] and for the case where n(r, A) is bounded [9], [10] and [11]. In the first part of this paper the case p = 0 is treated. The second part of the paper is devoted to the case 0 < p < (n 1) with the additional condition that the given analytic set A in Vis regular on an ( n 2 -p) -d imens iona l projective subplane of the infinite plane of the projective closure of V. The author wishes to thank Professor Wilhelm Stoll for his advice and encouragement.

Symmetric (Riemannian) spaces were introduced and developed by Cartan [1, 2] which led to the discovery of projectively symmetric (Riemannian) spaces by Soos [9]. Recently the theory of symmetric spaces has been extended to Finsler geometry by the present author [5]. The current paper deals with that class of Finsler spaces throughout which their projective curvature tensors possess vanishing covariant derivatives. Following Soos' terminology such spaces are calledprojectively symmetric Finsler spaces. Examples, conditions for a symmetric Finsler space to be projectively symmetric, reduction of various identities, and the discussion of a decomposed projectively symmetric Finsler space form the skeleton of the paper.

In [2] Knapp found necessary and sufficient conditions for two elliptic transformations to generate a discontinuous subgroup of L f(2, IR), the group of linear fractional transformations. In [6] Lyndon and Ullman give conditions for two hyperbolic transformations whose fixed points separate each other to generate a discrete free group of rank 2. In this paper we find necessary and sufficient conditions for the group generated by any pair A, B e L f ( 2 , IR) to be the discrete free product of the cyclic groups {A} and {B}. The results of this paper along with those of [2] resolve the question of when {A, B} is discrete in all cases except when the trace of the commutator a ( [A,B])= __2cos rrc, where r is rational. I would like to thank Professor Joseph Lehner for guiding me towards this problem. Also, the referee has kindly pointed out that Fricke and Klein discuss this problem as well as the case [A, Bl n-1 in [1].

As is well known there remain many unsolved problems in the CaIculus of Variations of multiple integrals; in particular, the theory for integrals in parametric form has not received much attention. This paper presents a canonical formalism with a new Hamiltonian function developed especially for application to the theory of geodesic fields of CARATHI~ODORY [2], or more precisely, to the form assumed by this theory for parametric integrals. As is pointed out by RUND [5], the canonical formalism given by CARATn~ODORY is not applicable to this case. Part one of the paper contains some brief immediate consequences of the parameter-invariance of the fundamental integral, and the definition and properties of the canonical variables and Hamiltonian function; these latter quantities being introduced by means of a certain contact transformation. In part two, some special features of geodesic fields of parametric integrals are pointed out, and an explicit characterization of geodesic fields by means of a generalized Hamilton-Jacobi equation is established. Finally a theorem concerning the integrability conditions of the field is proved, which restores an initial lack of determinism of the slope functions of a field.

This paper is concerned with two problems which, although not apparently closely related, are solved in part by the same methods. The first problem is: given a bicomplete (=comple te and cocomplete) category X and a triple T on X, is X T also bicomplete? The second is: given a category X and a functor R: X---~X, does R generate a free triple? This paper began as an attempt to show that the category of contramodules over a coring is cocomplete (see (4.4)). Many people, too numerous to mention, have contributed materially to the results and their applications. All notation and terminology not explicitly defined below may be found in the introduction to [-2]. The first section of this paper gives the main definitions used and in section two we give the fundamental lemma on which the proofs are based. The next two sections prove and give applications of the cocompleteness theorem. Section five gives the construction of free triples and in section six we apply this to show that if 3--~ is a small theory, then under certain conditions the category of ~-~ algebras in X is tripleable over X. In the next section we apply these results to the category of sets and we show that for a certain large full subcategory of endofunctors on sets there is a "free triple triple". The last section gives another cocompteteness theorem, not related to that of section three. This latter is a generalization of the result that every category of algebras over sets is cocomptete.

Section 1. Introduction Recent work of Conway [6, 7, 8] and of Shields and Rubel [21] has created new interest in the space C (S) of bounded continuous complex valued functions on a locally compact space S, given the fl or strict topology of Buck [4]. In both these papers some of the key proofs were made to depend upon certain results about the simpler space l~= C(N) of bounded complex sequences (N denotes the discrete positive integers). It is our purpose here to discuss rather thoroughly the space (C(S), fl), as a topological vector space, when S is discrete; when convenient or when it is necessary to emphasize that S is discrete we denote (as is fairly standard) this C(S) by I~176 For the sake of clarity and motivation, we indicate in Section 3 some of the known and needed facts about (C(S), fl), for S locally compact, including three results (based on constructions of Conway and of Shields and Rubel) which show how the space (l | fi) arises naturally in the study of general (C(S), fl) and its subspaces. Our basic results on (l~176 fl) and its subspaces may be found in Section 4, including both some new theorems and new proofs of several known results. The material here on the ew* topology and Lemma 4.2 is new and makes it possible to develop this section without using the rather complicated machinery of [8]. At the end of Section 4 a theorem is proved which says that (C(S), fl) is nuclear if and only if it is finite dimensional, a result which says in (l ~ (S), fl) that absolute and unconditional summability ace distinct when S is infinite. The final Section 5 is concerned with some special mappings into Co, with applications. A remark about this paper is perhaps in order: with respect to problems about Schauder bases (see [9]), the space (l ~176 fl) seems to play a role among non-metrisable, non-barrelled spaces analogous to that of the space Co among Banach spaces and thus may well be of some interest to those interested in bases. Indeed, the fact that (l ~ fi) is not barrelled makes essential the use of Conway's theorem that (1 ~~ fl) is a Mackey space (see Section 3), a result he derived from Schur's theorem that in 11, a(l 1, l ~~ and norm convergence for sequences is the same. This result, of course, is known to hold in the dual li(S) of (l ~~ (S), fl) [9, p. 33].

In this paper we consider some results on the orbits of groups of collineations, or, more generally, on the point and block classes of tactical decompositions, on symmetric balanced incomplete block designs (symmetric BIBD = (v, k, 2)system=finite 2-plane), and we consider generalizations to (not necessarily symmetric) BIBD and other combinatorial designs. The results are about the number of point and block classes (or orbits, i.e. sets of transitivity) and the numbers of elements in these classes. In Sections 2, 3 and 4 below we exhibit the key role of the rank of the incidence matrix of a design, while the remainder of the paper uses more specific properties of the incidence relations. Included in Section 2 is a simple new proof of the theorem of DEMBOWSKI [7] on the equality of the numbers of

If a finite group A of exponent 4 is the automorphism group of a torsionfree (abelian) group G, then A must be a subdirect product of cyclic groups C2 and C4 and of quaternion groups Qs (see [2, 4]); A is therefore contained in the variety V(Qs) generated by the quaternion group. Having dealt completely in [3] with the case of finite (or countable) abelian automorphism groups, we now turn to non-abelian groups (which, as we have mentioned before ([3], p. 34), must be of exponent 4 or 12). In this paper we confine ourselves to proving that for every n>2 the free n-generator group A, of V(Qs) is, in fact, the automorphism group of a suitably chosen torsionfree group 1. We also give at the end of the paper an example (with n = 3) of an epimorphic image of Aa of order 27 that cannot be an automorphism group, although it still satisfies the condition N3 of [2]: all its elements of order 2 lie in its centre. This group fails to satisfy the necessary condition N5 of [1] and cannot be a subdirect product of the kind described above.

Let A be a ring with an identity and let AJ~ be the category of all left Amodules; throughout this paper modules are assumed to be unitary. Let V be a finitely cogenerating, injective left A-module and ~(V) (resp. Y-(V)) the full subcategory of ad// consisting of all left A-modules X such that V-dora. dim X > 2 (resp. Horn A (X, V)--0). Then Y-(V) is a localizing subcategory of a~gg, and any localizing subcategory of A J{ is obtained in such a way. In a previous paper [7] we have proved that A./~/~- (V) ~ ~ (V) and that the ring which is constructed from A by Gabriel's process of localization with respect to ~(V) coincides with the double centralizer of V. Thus the subcategory N(V) of a~t r plays an important role in the theory of localization. The purpose of this paper is to give a characterization of N(V) as a full subcategory of A~g//. Our theorem in w 1 asserts that a full subcategory N of a~/~ is expressed as N(V) with a suitable V if and only if the following conditions are satisfied, where X and X' are left A-modules. (a) If X~-X ' and XsN, then X'EN. (b) IfXOX'eN, then X, X'eN. (c) If X~ for 2~A, then 1~ X~. ;,cA (d) XeN if and only if E(X)~N and E(E(X)/X)~N.

JAFFARD ([3], Chapt. t , Sect. 3) has shown that a directed partially ordered group may be imbedded, by means of an ideal extension, in a complete lattice semigroup. In this paper we show how this procedure may be generalised to a wide class of partially ordered semigroups, with, in addition, preservation of existing least upper bounds. In particular, we g~ve a construction which is applicable to a class of semigroups which includeS that of all residuated semigroups. Further, we obtain necessary and sufficient conditions under which a partially ordered semigroup m a y be imbedded in a conditionally complete group. We introduce in Section l the terminology used in the paper, and in the next Section the notion, of an ideal extension,-in terms of which we find necessary and sufficient conditions that a partially ordered semigroup may be imbedded, with preservation of least upper bounds, in a (conditionally) complete lattice semigroup. I t is shown that not every partially ordered semigroup may be so imbedded. We describe the result mentioned above for residuated semigroups, and complete the Section by proving some general results on ideal extensions. In Section 3 relations between different ideal systems are considered, and in the fiflal Section we establish necessary and sufficient conditions under which a partially ordered semigroup may be imbedded in a conditionally complete group.

Introduction The purpose of this paper is to study the Banach space Ho~ (G x G) of bounded analytic functions in a region G x G in C 2 by representing it as a space of bounded linear operators. By considering the compact operators we are able to relate H~ (G x G) and its subspace H~ (G)Q~H~ (G). (Definitions are given below.) In general, this subspace can be proper thereby providing an answer to a question raised by RossI during the Tulane Symposium on Function Algebras in 1965. We obtain a representation of H~o (G x G) as the dual space of a 7-tensor product of measures which is identified with M 1 (G x G) and LI(G x G)/N2. Furthermore, the compact linear operators from MI(G) to H~o (G) are characterized in terms of a 2-tensor product of L| spaces. By a constructive method we give a large class of functions on the unit polydisc D x D which are in H~ (D x D), but which are not in H~ (D) | (D). Finally, in Section 5, we summarize the relationships of all the Banach spaces considered in this paper. Obviously, we are strongly influenced by the papers of RtJBEL and SmELOS [5, 6]. They indicated the possibility of extending their results to higher dimensions. Such an extension to two complex dimensions is obtained when H~ (G x G) is exhibited as the dual of the separable Banach space M I(G x G). Using similar proofs and replacing arguments involving point masses by arguments based on approximate identities, many of our results can be proved independently of [5, 6] for the Banach space H~(TxT) of functions on the torus which are boundary values of bounded analytic functions on DxD. We wish to express our gratitude to Professor QtJIGLEY for many helpful discussions and to Professor STOUT for communicating some useful remarks.

w 1. Introduction In this paper, "group" is to mean finite group. We shall consider certain properties of groups admitting fixed-point-free automorphisms (that is, automorphisms which leave only the identity element fixed). Such groups have been studied off and on for many years. BURNSIDE proved before 1897 that a group admitting a fixed-point-free automorphism of period two is abelian, and FRo~ENItJS proved in 1901 that a group admitting one of period three is nilpotent of class at most two (see [2]). The conjecture of FROBENIUS that a group admitting a fixed-point-free automorphism of prime period is nilpotent was proved for solvable groups by WITT about 1936 (unpublished); that the class of nilpotence is bounded by a function of the period was proved by G. I~IGMAN [9] in 1957. The proof of the FROBENIUS conjecture was completed by THOMVSON [I4] in 1959, when he showed that a group admitting a fixedpoint-free automorphism of prime period is solvable. In 1961, GOPCE~STEIN and HERSTEIN [6] proved that a group admitting a fixed-point-free automorphism of period four is solvable and has a nilpotent commutator subgroup. The question of the solvability of groups admitting fixed-point-free automorphisms of arbitrary order has not been settled, and will not be studied here, where we shall assume the groups to be solvable. (Because of the theorem of THOMVSON and FEIT [4], that all groups of odd order are solvable, this restriction is not very severe.) The main result of this paper is a proof that, except possibly in certain special cases, a solvable group admitting a fixed-point-free automorphism of period pn, p a prime, has nilpotent height at most n (see p. 265). We now introduce some of the properties studied in this paper. Let G be a solvable group. Let V(G) be the maximal nilpotent normal subgroup of G. Because the product of two nilpotent normal subgroups is also a nilpotent normal subgroup, V(G) is a weU-defined characteristic subgroup of G. We let V 1 = V(G), and inductively define Vk SO that Vk/Vk_ 1 = V(G/V~_ 1)" V(G) is called the Fitting subgroup of G. If V, = G, the Fitting length or nilpotent height of G is =n. We note that if G and H are groups, V(G x H)= V(G) x V(H). Thus we see easily that the Fitting length of the direct product of two groups is the maximum of the Fitting lengths of the factors. We further note that if 1 c Ni N z ~... ~N~ = G is a characteristic series for G with nilpotent factor groups

In this paper we examine the geometrical structure of the orbits of coilineation groups of finite projective planes. In a recent paper, [6], FOVLSER and SANDI.ER have examined some of the combinatorial properties of these orbits. To do this they associated with each point orbit 9 the set of all lines incident with at least two points of t and looked at the numerical relationships which occurred. Our approach is to look at the configuration fixed by the subgroup stabilizing an orbit elementwiseo In [3] DEMBOWSKI showed that any collineation group F acts faithfully on at least one orbit, but that it need not act faithfully on both a line orbit and a point orbit. In w 2 and w 3, we show that the latter situation can occur only if the configuration fixed by F is a point p and a number of lines through p, or its dual (see Theorem 2.8 and Lemma 3.4). We complete w 3 by giving examples of such groups. In w 4 we consider quasiregular collineation groups. DEMBOWSKI and PIPER [5] have determined the orbit structure of all quasiregular collineation groups F with [ F [ > 89 (n 2 + n + 1), where n is the order of the plane on which F operates. This numerical restriction on I F] means that F has exactly one faithful point (or line) orbit. We investigate the orbit structure of F if F has exactly one faithful point (or line) orbit without putting any numerical restriction on IF I. We show that, unless the fixed configuration of F is a point p and a number of lines through p (or the dual), then the orbit structure of F is one of those determined in [5].

We shall be concerned in this paper with the derivatives of a class of functions f(z) which are analytic and univalent in the unit disc D: ]z[ < 1. It is known [2], for example, that the derivative f' (z) of a univalent function can have radial limits only on a set of measure zero on the circle K: ]z[ = 1. [Indeed, this class of univalent functions was constructed to settle a problem of Bloch and Nevanlinna ([-4], p. 138), namely, whether the derivative of a function of bounded characteristic is also of bounded characteristic. Since the derivative of a function in the class of univalent functions described above possesses radial limits on a set on K of measure zero, it is clear that the derivative of a function in this class cannot be of bounded characteristic, so that the Bloch-Nevanlinna problem has a negative answer. For Frostman's original solution, see [2].] However, it has been observed by McMillan that it is a trivial consequence of the Koebe Verzerrungssatz that the derivative f' (z) of any univalent function is a normal analytic function (cf. [1]), so that, by a theorem of MacLane [-3], f'(z) possesses angular limits at a set of points which is dense on K. Now the class of univalent functions which we shall examine in this paper, namely, the class of univalent functions whose derivatives possess radial limit values on a set of measure zero on K, will prove useful in demonstrating certain properties of the class of all univalent functions, as in Theorem 1, where we prove that the derivative of any univalent function possesses radial limits (and hence angular limits) at a non-denumerable set on K. If we compare the derivatives of this special class of univalent functions with the modular function, which is also normal in D, but has the property that the dense set on K at which it possesses angular limits is denumerable, we see that this special class of univalent functions is of independent interest. In particular, Theorem 3 provides a sufficient condition for the existence of radial limit values other than 0 and ~ for the derivative of a function of the class under consideration.

in which each term is normal in its successor and each factor satisfies either the maximum or the minimum condition for subgroups. A group G has the subnormal intersection property if the intersection of each collection of subnormal subgroups of G is itself a subnormal subgroup of G, that is, if G contains a subnormal closure of every subgroup of G. For this to happen it is sufficient, but not necessary, that the defects of the subnormal subgroups are bounded. In [6] Robinson shows that a finitely generated soluble group has the subnormal intersection property if and only if it is finite-by-nilpotent. Hence, in particular, soluble groups satisfying the maximum condition for subgroups and having the subnormal intersection property must be finite-by-nilpotent. On the other hand a soluble group satisfying the minimum condition for subgroups is an extension of a periodic radicable abelian group by a finite group, by a well-known result of Cernikov [2], and as such it has the subnormal intersection property (see, for instance, Lemma 2.2 of [7]). In this paper we show that a soluble minimax group with the subnormal intersection property is an extension of a radicable abelian group satisfying the minimum condition by a (torsion-free nilpotent)-by-finite group (Theorem A). However a soluble minimax group with this structure need not have the subnormal intersection property, as the infinite dihedral group shows. Thus Theorem A does not give a complete picture of the structure of soluble minimax groups having the subnormal intersection property. The structure of these groups is probably rather complex. However there is a subclass of the class of groups under consideration which is easy to characterise. A soluble minimax group whose every subgroup has the subnormal intersection property is an extension of a group satisfying the minimum condition by a nilpotent group, and conversely every subgroup of a group in the latter class has the subnormal intersection property (Theorem B). The material in this paper forms part of a Ph.D. thesis for the University of London. I wish to thank my supervisor Dr. D. J. S. Robinson for his valuable

The task of this paper is to classify the sets of conjugate matrices and to find a representative of a not too difficult shape in each such set. Throughout this paper we assume that K is contained in the field of complex numbers. An especially intensive treatment is being done for certain real fields K because real symplectic groups are important in the theory of SlEGEL'S modular functions. This paper was partially supported by an N.S.F. grant.

A subgroup V of G is called an j in jector of G if Vc~ N is F-maximal in N (that is, Vn N is a maximal ~-subgroup of N) for each subnormal subgroup N of G. It is known that the ~-injectors of a group G form a conjugacy class of subgroups 1,2]. If ~-injectors are always normal then ~ is called a normal Fitting class. In I-1] and [-3] some work has been done to classify normal Fitting classes. It has been shown in these papers that to any normal Fitting class there corresponds an object called a Fitting pair (f, A) defined as follows: A is an Abelian group and to each finite soluble group G there exists a homomorphism fG: G~A such that for all N ~ Gf~ = f~lN, where f~lN is the restriction of fG to N. Each such pair defines a Fitting class ~ given by G ~ if and only if fG(G)= 1, 1,Satz 3.1, 1.]. We use this to construct a Fitting class which contains $3, the symmetric group on 3 letters but does not contain the dihedral group of order 18. For some time it had been undecided whether the Fitting class generated by S 3 was the class consisting of all 3-groups extended by 2-groups. For a general survey see the paper of Cossey read at the conference on group theory held in Canberra, Australia 1973. The construction will actually provide uncountably many normal Fitting classes for all of which, the injectors have index 2. We will define and prove the existence of these classes by using Fitting pairs. Let f2 be the set of powers of odd primes and let 0 E f2. Let A be the cyclic group of order 2 and define, for any finite soluble group G

Dr. G.L.C. Bond of the University of Reading has pointed out a mistake in my above paper. In the proof of Lemma 3.2 I made the inadvertent assumption that all rings are commutative, and in fact the lemma fails for the ring of 2 x 2 upper triangular matrices over Zp, p prime. Fortunately the error can be confined to Lemma 3.2, as was also pointed out by Dr. Bond. The only place where 3.2 is used is in the proof of 3.3, and here its use can be avoided as follows: we have R / J ~ T 1 G " "• 7", where each T/~292, and we pick elements ejeR which map by the canonical homomorph i sm R ~ R / J to the identity of T i. We find that e , = x + a where x e J and aeAnn(J ) . At this point we wish to assert that e,ZsAnn(J) for some integer 2>0 . In the paper we invoked 3.2; instead we show that )~ = c 1 will do, where c is the index of nilpotency of J. For then c-1 =(x+a)C-1 en

where IKP(oo) denotes the oo dimensional (left) IK projective space, d=dim~aIK, and of, r( ) denotes the framed bordism homology theory. In this note we develop some elementary properties of S~:, and show how the Adams-Toda complex e invariant er 2]~Q/7Z can be computed in terms of 2 $ M. We then apply these results to finite cyclic coverings of framed manifolds coming from principal circle actions. This is followed by applications to homogeneous spaces of compact Lie groups. After indicating the modifications needed for the quaternionic case we close the paper with some results indicative of the more general theory of equivariant framed manifolds which we develop in a sequel. It is a pleasure to thank the Mathematische Institut at Oberwolfach for their kind hospitality during the topology conference in September 1973 when this research was begun.

A differentiable function f in a field with a non-archimedean valuation K possesses the following "quasilinearity property": Denote by lal the valuation of an element a of K and assume that f has at a point x o a non-vanishing derivative f'(Xo). Then for all x in a neighbourhood of x o I f (x ) f (Xo) l = ]f'(Xo)[ IX-Xo[. In a short note I-5] the author published without proof some theorems based on this property and introduced the relevant concepts radius of linearity and radius of similarity of a function at a point. Since interest in non-archimedic analysis has recently awakened again [3] and since 1-5] is not easily accessible, some properties of differentiable mappings are developed more fully in this paper, in c o , u n c t i o n with some applications. In addition to a stronger formulation of the basic theorem of [5], this article contains the following new results: 1) Conditions are set up under which differentiable mappings are similarity mappings locally (w 4, Ths. 3, 4). 2) The proof of a theorem on the differentiability of a function series, first cited in [5], is presented (w 5, Th. 5). The conditions include data on the linearity radii of the series (renamed quasilinearity radii in this article). 3) In a new group of theorems the quasilinearity radius of an infinite function series is determined in terms of the quasilinearity radii of the terms or of the partial sums of the series (w 5, Ths. 5, 6). 4) The quasilinearity radii of the particular functions x z, x 3, sin x, cos x, and e x are calculated and those of x" (for general n~N) and of the functions

Section 1 In the literature of the spectral theory of linear operators there are many results concerned with the problem of determining the essential spectrum of selfadjoint operators associated with these differential operators. This paper considers a related and in some sense more delicate problem. Namely, how can one tell if the spectrum of the operator to the left of a given real number # is finite? While it is known that the essential spectrum is invariant under a wide variety of perturbations, it still seems to be unknown whether the finiteness property enjoys a similar invariance. One general finiteness criterion due to Hartman and Putnam [5] deals with the case of the ordinary differential operator l[y] = -y"+q(t)y, t>=O. It was shown that if the operator is in limit-point case the spectrum of any selfadjoint operator associated with I is finite to the left of # if the equation l[y] = #y is non-oscillatory. This condition means that there is a solution to l[y]=#y which is positive for all sufficiently large t. In [7] the author extended this result to a class of formally selfadjoint second order elliptic operators L

Let G be a finite group with BN pair (in the sense of Tits [10]). Following the terminology of Green [-6], the irreducible complex characters of G which are components of the induced character (18) G are called the Steinberg characters of G. In [6] Green has proved that if ~ is a simple Lie algebra over the complex field and if ~(q) is the corresponding Chevalley group (see [-3]) defined over the field with q elements, then for q sufficiently large the degrees of the Steinberg characters (other than the 1-character) of ~(q) are divisible by p, the characteristic of GF(q). The purpose of this paper is to determine precisely how large q has to be. In view of the results obtained by Seitz [-8] this permits the doubly transitive permutation representations of most of these groups to be classified. It is hoped soon to treat the groups of twisted type in a similar fashion. I am deeply indebted to Dr. R.J. Clarke for invaluable discussion.

In recent years it has become popular to use sheaf spa :es to obtain representation theorems and/or embedding theorems for various alge- bras-for example: semigroups (Adams [1], Keimel [15, 17]), rings (Davis [9], Dauns and Hofmann [6], Hofmann [12], Kist [21], Mew- born [22], Mulvey [24], Peercy [25], Pierce [26]),/-groups (Davis [8], Keimel [16, 18]), f-rings (Keimel [13, 14, 16, 18]), distributive lattices (Davey [7]), and for universal algebras (Comer [3, 4], Keimel [15], Keimel and Werner [19]). See also the extensive list of references in [12]. Thus it is natural to investigate to what extent the various constructions used concur. We carry out our investigation in four parts. In Section 1 we present the basic properties of sheaf spaces and sheaves of universal algebras. In Section 2 we give a general procedure for the conversion of a subdirect product representation of an algebra into a representation as an algebra of global sections of a sheaf space; this generalizes the procedure used in all of the papers mentioned above. Section 3 gives a brief study of the elementary properties of sheaf spaces and sheaves over Boolean spaces, and in Section 4 we apply our results to construct and study the extension FA of an algebra A induced by a Boolean algebra of congruences. The constructions of [1, 7, 8, 9, 14, 17, 21], and [25] then arise as particular cases. Under certain conditions the extension is in fact an isomorphism, in which case the general representation theorem of Comer [3] is obtained. Finally, we prove that F A has a (choice free) construction as a direct limit, showing that each of the constructions listed above may be obtained in a manner analogous to that for/-groups in Conrad [5] and for semiprime rings in Speed [28]. We use standard universal algebra terminology. The lattice of congruences on an algebra A is denoted by I$(A), with least element co and greatest element ~. If

In this paper the concept of distributivity introduced earlier [1] is used to show that a homomorphism with respect to two distributive operations which is extended as a homomorphism with respect to operation 1 remains necessarily also a homomorphism with respect to operation 2 on the 1-closure of the original domain of definition. The result is illustrated by applications to continuous extensions of homomorphisms between δ-complete vector lattices, association of families of stochastically independent systems of sets and integration of products of independent functions.

A subgroup of a group G is called quasinormal in G if it permutes with every subgroup of G. For a finite group G it is well known that quasinormal subgroups are not far from being normal: It6 and Sz6p [3] have shown that a quasinormal subgroup of G containing no nontrivial normal subgroup of G is always nilpotent. In this paper we shall prove a stronger result, namely: Theorem. If Q is a quasinormal subgroup of the finite group G, then Q~/QG is contained in the hypercentre Zoo (G/Q~) of G/Q~.

In [2] Lawvere characterized the category of sets by elementary axioms. using a language, with one sort of variable symbols (mappings) and two unary functions symbols (domain and codomain 1) and one ternary relation. composition. A r >B means f is a map with domain A and codomain B. Schlomiuk [5], presented a method of characterizing the category of topological spaces by using the full subcategory of discrete spaces; the fact that the functor, inclusion, from sets to topological spaces has a left adjoint, together with additional axioms on a certain constant ~the two-point space ({a, b}, {a}, {a, b}). Lawvere also characterized algebraic categories [1], using the special properties of a certain constant, the free object on one generator. and an adjoinmess condition. It is conjectured that what one needs is a previously characterized reflective subcategory cg and finitely many objects at . . .a . determined up to isomorphism, such that the full subcategory on cg u ~ ai, is in the transitive closure of" adequate". We shall adopt this technique i in the present paper to the category of relational systems~

Pietsch [5] introduced the concept of absolutely summing operators in Banach spaces and later in [6] extended this concept to absolutely p-summing operators. At the background of these concepts are the sequence spaces I p and their duality theory. The object of the present paper is to extend the above concept to abstract sequence spaces 2. The sequence spaces 2 involved are described in Section 2; the absolutely 2-summing operators are studied in Section3 while Section4 discusses the interesting special case 2=n(~b), a sequence space which includes for special q5 the P and l ~ spaces and was introduced in the literature by Sargent [10].

The purpose of this paper is to investigate a situation which is best described by the following examples. I. Call a connected, locally connected and locally simply" connected topological space with base-point perfectly connected in brief. By a covering we mean a base-point preserving, surjective local homeomorphism of a perfectly connected topological space into a like one. Let ~0 be a ut,iversal covering of e'. Then 9 plays a double r61e. 1 ~ To each base-point preserving, continuous mapping ~0' of a simply and perfectly connected topological space into e', there exists one and only one base-point preserving, continuous function [ rendering the diagram

In 1938, Wielandt announced results, [7-1, on the structure of (what are now called) 2-primitive near-rings with identity and finiteness conditions. Laxton, [5], independently classified finite, 2-primitive distributively generated near-rings. His methods may be extended to obtain Wielandt's results in the more general situation, see for example [6] and [3]. However, Wielandt's proof ([2]) makes use of an interesting lemma, which gives rise to the density theorem for 2-primitive near-rings with identity. If we turn our attention to the structure of 0-primitive near-rings with identity, we do not, so far, possess a full density theorem; however, some of Wielandt's work can be adapted to the more general case, and gives us a partial density theorem, which is, in fact, of some use. This paper examines an interesting subclass of the class of 0-primitive near-rings with identity and d.c.c, on right ideals. I would like to thank E.W. Wallace for his careful supervision while I was studying for the degree of Ph.D. at Leeds University. These results form part of my thesis. The Science Research Council supported me with a Research Studentship during this period. I am grateful to G. Betsch and H. Wielandt for making available to me some of their work, and to G. Betsch for some helpful suggestions.

In a preceding paper [8], we studied a new method of proving the existence of solutions of nonlinear functional equations involving a densely defined monotone operator T from a reflexive Banach space X to its conjugate space X*. This method, which uses approximating equations containing compact monotone operators in an auxiliary Hilbert space, we called elliptic superregularization because an important special case of its application consists of treating variational boundary value problems for quasi-linear elliptic differential equations of order 2m (m> 1), in the form

Some steps have been taken towards showing that, if G is a finite group and V is a fixed-point-free group of automorphisms of G such that either V is cyclic or (IVI, 1G[)= 1, then G must be soluble. (See, for example, [3-6] and [2], Chapter 10.) In this paper we do two things. In Section 2 we derive some information about a minimal counterexample to a conjecture of this sort, and in Section 3 we use that information to obtain a proof of the following.

The general problem, with a particular instance of which the present paper is concerned, is to obtain a description of the local structure of a group from information about the global structure. The aspect of local structure investigated here is the embedding of subgroups, especially of nilpotent subgroups in finite soluble groups. A classification of embeddings of subgroups in finite groups by means of an arithmetic function called abnormal depth was proposed in [6]. Let H be a subgroup of a finite group G. Then a(G:H), the abnormal depth of H in G, is the least number of abnormal links appearing in any balanced chain of subgroups connecting H to G, that is a chain for which each link is either normal or abnormal. Thus a (G:H)= 0 if and only if H is subnormal in G; and a(G:P)__< 1 for every subgroup P of G of prime power order. It was shown in [6] that if H is a nilpotent subgroup of a finite soluble group G, of nilpotent length n, then a (G: H) =< n - 1. Here in w 1 we examine in greater detail the easiest non-trivial case, in which n = 2, and then in w 2 prove certain supplementary results for n = 3 and n = 4. Some simple wreath product properties are established in w 3 and used in w 4 for the construction of examples showing that the embedding results obtained cannot be improved in various obvious ways. Notation and terminology follow common usage. If t; and ~ are classes of groups, then 3s ~ denotes the class of all groups G having a normal subgroup X such that X e 3~ and G/X e ~. This defines a composition of classes of groups which in general is not associative. However, we shall deal only with classes of which the composition is associatNe, and we may therefore omit brackets from products of more than two classes. Since we shall be concerned exclusively with finite groups, we take 91 to denote the class of finite nilpotent groups and 9.1 the class of finite abelian groups. Then for any positive integer n, 9l" is the class of finite soluble groups of nilpotent lengths

Let K be a local field in the unequal characteristic case and let k be the residue-class field (char k = p > 0). In this paper we want to show that there is a connection between the cohomological p-dimension of the Galois group G K of K and the p-degree of the residue-class field k (i. e. the number of elements in a p-basis of k over kV). We first show that the cohomological q-dimension of G r for any prime q depends only on the residue-class field k, using a decomposition theorem which tells that the canonical exact sequence of Galois groups

The group G of the title is obtained as a primitive permutation group of degree 100 in which the stabilizer of a point has orbits of lengths 1, 22 and 77 and is isomorphic to the Mathieu group M22. Thus G has rank 3 in the sense of [1]. G is an automorphism group of a graph constructed from the Steiner system ~ (3, 6, 22). WITT [3] defined a Steiner system ~(d, rn, n) to be a set S of n points together with a set B of subsets of S (referred to here as blocks) such that each block contains exactly m points and each set of d points is contained in exactly one block. WITT [4] showed that Steiner systems ~ (3, 6, 22) exist and that they are unique up to isomorphism. The automorphism group Mz2 of an ~ (3, 6, 22) contains the Mathieu group Mz2 as a subgroup of index 2 and is the normalizer of M22 in M24. Throughout the rest of the paper we shall use the following notation: S and B will denote the sets of points and blocks, respectively, of a fixed ~(3, 6, 22). Points will be denoted by Greek letters ~, fl, ... and blocks by Roman letters u, v, .... For each o~eS, [~] will denote the set of blocks containing ~. We shall use the following facts about ~(3, 6, 22) and M22

Finite groups with abelian Sylow 2-subgroups have been classified by Walter [8]. In this note I want to describe an alternate proof of some partial result of Walter's work, namely the theorem stated below. It represents the first major reduction step in that classification. The approach used here is to some extent derived from [1]. ! Besides the groups L 2 (q)= PSL(2, q) another class of simple groups enters our discussion: We say that a simple group G with abelian Sz-subgroups is of type JR (Janko-Ree) if, for any involution t in G, CG (t) is a maximal subgroup of G isomorphic to ( t ) | E where PSL(2, q)~ E ~_ PFL(2, q) with odd q > 5. In fact, E = L 2 (q), as proved by Walter 1-7] ; and the structure of G is very well known by results of Janko, Thompson, and Ward, see [4-6] and [9]. Unpublished work of Thompson is very close to a complete identification of such a group G. For further information about the subject see the introduction of 1-8] and 1-3; 16.6]. A group G with abelian S2-subgroups is an A*-group if G has a normal series 1 _~ N_~ M _~ G such that N and G/M are (solvable) of odd order and M/N is a direct product of a 2-group and simple groups of type L 2 (q) or JR. Note that the result of Walter mentioned above means that an A*-group is an A-group (in the sense of 1-3; 16.6]). The rank r(X) of a group X is defined in the following way: X has an elementary abelian p-subgroup of order p,(X), for a suitable prime p, but none of order prtX)+1, for any prime p. Throughout this paper, "g roup" means "finite group".

For a non-commutative ring there exists not always the classical quotient ring, while there exists always the maximal quotient ring. In the case of a semiprime ring, howevel; to have a semi-simple Artinian maximal quotient ring implies the existence of the classical quotient ring (cf. Johnson [6, 4.4 Theorem] and Sandomierski [17, Theorem 1.6]). Therefore avoiding the use of the Ore condition it may be possible to find some conditions, under which the existence of the classical quotient ring is assured by requiring the chain condition on a ring which is constructed by a suitable localization. The purpose of the present paper is to provide such conditions and localizations. For a ring with an identity every idempotent topologizing filter ~is defined by a finitely cogenerating injective right R-module W in the following way: ~ = {right ideal D of R I HomR (R/D, W) = 0}, and we have the localizing functor H such that H(X)=~ Horn R (D, X/X~), where X is a right R-module DE~ and X ~ = {xeXIAnnR x e ~ } . H(R) has a ring-structure and H(X) becomes a right H(R)-module in a natural way. Recently, Morita [14] has shown that H(R) and the double centralizer Q of W R are identical and H is exact. With these notations we can state our main theorem in w 3: The following statements I to III are equivalent.

By the rank of a transitive permutation group we mean the number of orbits of the stabilizer of a point thus rank 2 means multiple transitivity. Interest is drawn to the simply transitive groups of "small" rank > 2 by the fact that every known finite simple group admits a representation as a primitive group of rank at most 5 while not all of these groups have doubly transitive representations. In this paper we consider finite transitive groups of rank 3, a class of groups which seems to have received little direct attention.

The theory of primal algebras has recently been generalized in several ways. In particular the concept has been extended on the one hand by the admission of subalgebras and internal isomorphisms to the successively larger classes of semi-primal, infra-primal and quasi-primal algebras [1, 3, 8], and on the other hand by the admission of proper congruences to the hemi-primal algebras [-23. In [53 the authors introduced the concept of "total" algebras: finite algebras in which all mappings which preserve all subalgebras, automorphisms and congruences are representable by polynomials. So defined, this class of algebras was shown to comprehend both the infra-primal and hemi-primal algebras, but, on the other hand, it is easy to see that the quasi-primal algebras (since they may possess non-identical isomorphic subalgebras) are not always total. In the present paper we enlarge the class of total algebras to include the quasi-primals as well as the semi-, infra-, and hemi-primals. Our principal focus is on systems of weakly independent quasi-primal algebras, their products, and the varieties generated by them.

By a choice mapping we mean a functio~ defined on some class of sets mapping a set of that class on some subset or possibly on some superset, Let Q be the class of all fimte sets of distinct positive integers. Let each set be arranged in a monotonically increasing order. Define by O' the subclass of Q of sets possessing more than one, elemenl= Let ~r = (al, .." , a~) be an m'bitrary element of O', It is shov~Tt subsequently in this paper that there is precisely one positive integer io, such that for att i, t