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The first part of the theorem of the above paper states that if ois a closed convex set in real X1 ... Xn-space which is invariant under permutations of coordinates, and if C(o-) denotes the set of real symmetric nXn matrices whose eigenvalues X1, X. form the coordinates of points in a, then C(o) is convex. I am obliged to Professor R. T. Rockafellar for pointing out that this statement is essentially contained in a theorem of Chandler Davis.2 The statement also follows from an earlier result of V. B. Lidskii,3 which was not known to me at the time of publication.

A p-ring (or generalized Boolean ring) P is a ring of fixed prime characteristic p in which ax=a for all a in P. In this paper P is partially ordered by a relation which is a generalization of the usual Boolean order. A subset S of P is then called quasiorthogonal if ab(a-b)=O for all a, b in S. It is shown that P is injective in the category of p-rings if and only if every quasiorthogonal subset has a supremum under this partial order. Sikorski [4] has shown that in the category 9 of Boolean rings the injective objects are the complete Boolean rings. The purpose of this paper is to present a generalization of this result to the category 9 of p-rings, where 9 is understood to be the category with objects rings P of fixed prime characteristic p in which aP=a for all a e P and with morphisms the usual ring homomorphisms. If P is a p-ring, then the set B(P) of idempotents of P is a Boolean ring under the multiplication of P and the new addition defined by aE3b=a+ b-2ab. Batbedat [2] has used B to establish an isomorphism between b? and R. (See also Stringall [6].) Hence, the injective objects in 9 are simply those P for which B(P) is a complete Boolean ring. In this paper the irnjective objects in b? will be characterized by a kind of completeness of a particular partial order that is an extension of the usual partial order on the Boolean ring of idempotents. DEFINITION ]. For a, b E P, a_b if and only if aP-1b=a. It is easily shown that _ is a partial order on P (see, e.g., Abian [1]), but in general (P, _) is not a lattice. It is, however, a lower semilattice with meet defined by aAb=a-a(a-b)P-1. A simple calculation shows that a(bAc)=abAac and aAO=O for a, b, c e P. Foster [3] has shown that every element a of P is uniquely expressible as a sum of elements {ej(a): jE Z.} of B(P). In particular, v-1 (1) a = Zjej(a) i=l Presented to the Society, January 26, 1973 under the title Injectivity in the category of p-rings; received by the editors February 27, 1973. AMS (MOS) subject classifications (1970). Primary 06A70; Secondary 06A40, 06A23.

In the author's paper [1 ] it is shown that if a sequence of orientable n-dimensional generalized closed manifolds (abbreviated n-gcm) converge (n-1)-regularly to an n-dimensional set, then the limit set is also an orientable n-gcm (see Definitions 1 and 2). In this paper a similar result is obtained for manifolds with boundary. Throughout the paper we assume that our sets are imbedded in a compact Hausdorff space S and that the cycles are Cech cycles with coefficients in an arbitrary field which we will omit from our notation for a cycle. All of the basic homology theory needed is in [7] and a knowledge of it will be assumed. We shall use the notation { A } --+A to mean that the sequence of sets A iCS converges to A CS as a limit (see p. 10 of [5]). We shall consider convergence only when all of the sets Ai are closed, and shall not explicitly state this henceforth; the limit set, as is well known, will always be closed. We shall use small latin letters for points, large ones for sets, and script letters for finite collections of open sets which are coverings of S. We shall use "U" for point set union or sum, "n" for intersection, reserving + and - for the group operations. By VAA we shall mean the subcomplex of the nerve of V whose vertices are elements of the covering V that have a nonvacuous intersection with A. If every element of V is contained in some element of V, we shall say that V is a refinement of V, and shall write V >V, or V = V(V); we shall use the notation' IIH. to denote a simplicial projection from the nerve of V into the nerve of V. By U C C V we mean that UC V, and by V>? we mean that the closure of the elements of V form a refinement of V, and by V> *V, we mean that the stars of the elements of V form a refinement ofV. If a cell a of a covering V has a nucleus (the intersection of all the open sets representing the vertices of a) that meets a set B, we say that af is on B; if all the cells with nonzero coefficients in a chain Cr ofZ V are on B, we say that Cr is on B. If the nucleus of a is in the open set Q, we say a is in Q. We shall use

Let d be a divisor of a positive integer n. Then d is a unitary divisor if d and nld are relatively prime, and d is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. An integer is bi-unitaty perfect if it equals the sum of its proper biunitary divisors. The purpose of this paper is to show that there are only three bi-unitary perfect numbers, namely 6, 60 and 90. A divisor d of an integer n is a unitary divisor if d and nld are relatively prime. A divisor d of an integer n is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. Let a(n) be the sum of the divisors of n, let ?*(n) be the sum of the unitary divisors of n, and let ?**(n) be the sum of the bi-unitary divisors of n. We say that N is unitary perfect if a * (N) = 2N. Subbarao and Warren [2] showed that 6, 60, 90 and 87360 are the first four unitary perfect numbers; Wall reported [3] that 146,361,946,186,458,562,560,000 = 2183.547.11.13.19.37.79.109.157.313 is also unitary perfect and later showed [4] it to be the next such number after 87360. Subbarao [1] has conjectured that there are only finitely many unitary perfect numbers. We say that N is bi-unitary perfect if a* * (N) = 2N. The purpose of this paper is to show that the first three unitary perfect numbers, i.e., 6, 60 and 90, are the only bi-unitary perfect numbers. One easily verifies that ov* is multiplicative and that if p is prime and e> 1, then rY**(pe) = ry(pe) = (pe+l l)/(p 1) if e is odd, and or**(pe) = (pe+l l)/(p 1) -pe/2 if e is even. Hence a**(n)?a(n) with equality if and only if every prime which divides n does so an odd number of times. It also follows immediately that a* * (n) is odd if and only if n is 1 or a power of 2; consequently, each odd prime power unitary divisor of n contributes at least one factor 2 to a**(n). Received by the editors June 10, 1971. AMS 1970 subject classifications. Primary lOA20; Secondary lOA99.

In a previous paper [7], we studied the structure of HID spaces. In this paper, we consider the behavior of HID spaces under monotone mappings. The principal result of this paper is that, given an arbitrary compact metric space Y, there is an HID space X and a monotone map f: X-) Y. We also show that an arbitrary HID space can be mapped monotonically onto a space of any preassigned dimension, and that, given an HID space X and a positive integer n, there is an n-dimensional space Y and a monotone map f: Y->X. R. H. Bing showed in [2 ] that each nondegenerate monotone image of a pseudo-arc is a pseudo-arc. The results of this paper show that no similar monotone invariance property holds for spaces of dimension greater than 1. In this paper, all spaces will be compact metric spaces (compacta). We will be dealing with the Hilbert cube, which we regard as being the product of a countably infinite collection of straight line intervals IX = 11 X 12 X 13 X ... , where Ij = [-1/2i, 1/2'].

In a recent paper the first author gave an explicit construction of a sequence of polynomials having their zeros on the unit circumference which converge boundedly to a given bounded zero-free analytic function in the unit disk. In this paper we find the best possible uniform bound for such approximating polynomials and construct a sequence for which this bound is attained. The method is also applied to approximation of an analytic function in the unit disk by rational functions whose poles lie on the unit circumference. Some open problems are discussed. Let C denote the unit circumference in the z-plane and D its interior. The term C-polynomial shall mean a polynomial all of whose zeros lie on C. In [6] the first author unified the results of MacLane [5] and Chui [2 ] by proving THEOREM A. Let f(z) = 1 +c+c2z 2+ * * * be a zero-free holomorphic function in D. Then there exists a sequence of C-polynomials Pn(z) assuming the value one at z =0 which converges to f(z) uniformly on every closed subset of D. If in addition M-supizi

In this paper a stronger form of aposyndesis is defined and continua with this property (strongly aposyndetic) are shown to be countable-set aposyndetic. Although every continuum which is a product of nondegenerate continua is countable-set aposyndetic, it is established that no product of nondegenerate continua is strongly aposyndetic. Throughout this paper a continuum is a compact connected metric space. Let M be a continuum. If N is a subcontinuum of M, the interior of N in M will be denoted by int N and the boundary of N in M by Bd N.2 A continuum M is aposyndetic at a point p if for each point q in M{p} there is a subcontinuum N in the complement of {q} such that peint N [3]. If the {q} above is replaced by a finite (countable closed) set then M is said to be finitely (countable-set) aposyndetic at p. A continuum is aposyndetic (finitely aposyndetic) (countable-set aposyndetic) if it has the given property at each point [7]. This familiar property can be strengthened as follows. DEFINITION. A decomposable continuum M is strongly aposyndetic provided that for any pair of proper subcontinua H and K such that M=HUK, each of H and K is aposyndetic. It is easily seen that each strongly aposyndetic continuum is aposyndetic and there are numerous elementary examples of aposyndetic continua which fail to be strongly aposyndetic. The following two theorems established sufficient conditions for subcontinua of a strongly aposyndetic continuum to be aposyndetic. The first theorem is an immediate consequence of the definition. THEOREM 1. If A is a nonseparating subcontinuum of a strongly aposyndetic continuum M and int A $ 0, then A is aposyndetic. Received by the editors October 15, 1970. AMS 1970 subject classifications. Primary 54F20, 54F15; Secondary 54B10.

In this paper it is shown that a commutative semisimple ring is isomorphic to a direct product of fields if and only if it is hyperatomic and orthogonally complete. In this paper we give a necessary and sufficient condition for a commutative semisimple ring R (i.e., R has no nonzero nilpotent element) to be isomorphic to a direct product of fields. In particular, we show that hyperatomicity and orthogonal completeness is such a necessary and sufficient condition. It is well known that without these conditions R is isomorphic to a subring of a direct product of fields [1, p. 16]. We would like to emphasize that, in what follows, R stands for a commutative ring with no nonzero nilpotent element. Thus, in particular, for every element x of R, (1) X2 =O if and only if x = O. We first prove several lemmas. Lemma 1 below, generalizes the corresponding result for Boolean Rings [2, p. 154]. LEMMA 1. The ring R is partially ordered by : where for every element x and y of R, (2) x 5 y if and only if xy = x2. PROOF. Since xx=x2 it follows from (2) that x gx. Thus, ? is reflexive. Moreover, if x9y and y

In this paper an example is given of a ring with left global dimension 2 having the property that the singular submodule of any R-module A is a direct summand of A. Although the example given is quite specific, the methods can be used to construct a fairly large class of these rings. In this paper, all rings are assumed to be associative with an identity element, and all modules will be unitary left modules. An R-module A is said to split if the singular submodule, Z(RA), is a direct summand of A. R is called a splitting ring if every R-module splits (see [1], [3], and [7]). In [3] Cateforis and Sandomierski have shown that every commutative splitting ring has left global dimension _1. M. L. Teply [7] has shown that if the commutative hypothesis is dropped, then every splitting ring must have left global dimension

In this paper a theory of pseudo-complements is developed for posets (partially ordered sets). The concepts of ideal and semi-ideal are introduced for posets and a few results about them are obtained. These results together with known results about pseudo-complements in distributive lattices lead to the main results. It is proved that if in a pseudo-complemented semilattice or dual semilattice every element is normal, then it is a Boolean algebra. Using this result new proofs for two known theorems are obtained. The existence of maximal ideals in posets is established and it is shown that the dual ideal of dense elements of a poset with 0 is the product of all the maximal dual ideals. Already, there exists a theory of pseudo-complements for lattices. Frink [5 ] has obtained a generalisation of the theory for semilattices. In this paper we extend some of the results of Frink [5] and Balachandran [1] to posets (partially ordered sets). We obtain these extensions by using the concept of semi-ideal, which we define in ?2. This paper consists of three sections. In ?1 we summarise some known results which we use in later sections. ?2 deals with some of the properties of semi-ideals and ideals in posets. Our definition of poset ideal is different from that introduced by Frink [4]; however in a lattice our definition is equivalent to the usual definition. Using the results obtained in ?2, we develop a theory of pseudo-complements for posets in ?3. 1. Preliminaries. We shall denote the ordering relation in a poset by ieI ai and rliET ai respectively. When A is finite, say, A = { a,, a2, * , an }, the lattice-sum and the lattice-product of the as are denoted by al+a2+ * . . +a. and a,*. a. respectively. The Received by the editors February 3, 1970. AMS 1969 subject classifications. Primary 0620; Secondary 0630, 0635.

In this paper, "point," "line," and "incident upon," are undefined terms. The phrases, "point is on a line," "point is incident upon a line," "line is incident upon a point," "line is on a point," are all to be considered synonymous. We will say that a line I intersects a line k (at a point P) if and only if P is on both I and k. Two lines are parallel if and only if there is no point which is on both. The following axioms will be used: AXIOM I. If P and Q are distinct points, there is exactly one line on P and on Q. AXIOM II. If I is a line, there is at least one point not on 1. AXIOM III. If I is a line and P is a point not on 1, there are exactly m distinct lines on P (m ?2) which are parallel to 1. AXIOM IV. There is at least one line with exactly n points on it, n _2. The entire set of points and lines whose existence is postulated by these axioms (for given m and n) will be called a geometry. Other work done with an axiom system containing the Axiom III as here stated is not known to this investigator, but Szamkolowicz [3 ] has reported on an equivalent system, and similar systems have been studied [2], [4]. As they are here stated, the axioms may or may not be consistent, depending on the values assigned to m and n. For n =2, their consistency for any m is established by the existence of a model constructed by A. N. Milgram [4], and for n = 3, m = 4, their consistency is shown by a model described by Abraham Barshop [I ]. This paper will demonstrate their incon sistency for m=2 and n>2.

R becomes a topological ring by defining the neighborhoods of zero to be the ideal (q) and all powers of (q). R is a complete discrete valuation ring if it is complete with respect to this topology. A non-Archimedean discrete valuation V can be defined on R by letting V(0) = co and V(a) (a^O) be the highest power of q to divide a. This valuation can be extended in the natural way to the quotient field K of R. The nonzero elements of R are then the quantities in K with non-negative valuation. Conversely, given a field K with nonArchimedean discrete valuation, the elements in K with non-negative valuation are the nonzero elements of a discrete valuation ring R, the ring of integers of K. R is complete if and only if K is complete. Again K is the quotient field of R. It is largely in this context that complete discrete valuation rings have been studied. It has been shown that if R is complete and has the same characteristic as its residue class field F then R is isomorphic to the ring of power series F[[x]]. Moreover, in the remaining case (R has characteristic infinity and F has characteristic p),Ris uniquely determined by F if R is unramifiied, i.e., if p is prime in R. Also there exists an R for any given F. If V(p) =n, »> 1, R is said to be ramified with ramification index n. For references see [2]. This paper is concerned with some aspects of the structure of ramified complete discrete valuation rings. Throughout this paper, the symbols R, R', Rn etc., will denote complete discrete valuation rings of characteristic zero having the same residue field F of characteristic p. The subscript on the ring symbol will designate the ramification index or if there is none the ring is unramified. Rings Rn have been studied extensively, largely as a part of algebraic number theory. Thus, as indicated below, a number of the results of this paper are known, however the methods are new and simple. Theorem 1 provides a description of an arbitrary ring Rn in terms of the unique unramified R and is closely related to the theorem [3, p. 237, Theorem 11 ] which states that every Rn is an R(t) where t is a root of an Eisenstein equation. This characterization is then used

In a recent paper, Felix E. Browder discussed continuous self-mappings on a metric space, satisfying a func- tional inequality. Browder gave sufficient conditions such that the successive approximations of any point for such mappings con- verge to a unique fixed point. In the present paper, Browder's result is extended to a commutative semigroup of mappings and also to single mappings that are not necessarily continuous and satisfy a weaker form of the functional inequality. 1. It is the purpose of this paper to generalize the following result of Felix Browder (2), under the assumption that the set M there is closed. Theorem 1. Let (X, d) be a complete metric space, M a bounded subset of X, Ta mapping of M into M. Suppose there exists a monotone nondecreasing function \p(r) for r = 0 with \f/ continuous on the right, such that if/(r) 0, while for x, y

Fell has proved that the process of inducing representations of a locally compact group from representations of closed subgroups is a continuous process if topologies are defined on the spaces of representations in the right way. As a corollary he shows that inducing preserves weak containment. This paper generalizes Fell's results to twisted group algebras. These algebras generalize the idea of the group algebra of a group extension, and the concept of induced representation extends in a natural way. We show that Fell's results will hold if the "cocycle pair" defining the twisting of the algebra is sufficiently continuous. Preliminaries. In [4] and [5], Fell discussed weak containment properties of induced representations of groups, and in particular he discussed the continuity of the inducing construction with respect to two topologies related to weak containment. Group algebras of group extensions are special cases of twisted group algebras as defined in [1], and in [1 ] the concept of induced representation was extended to these algebras. It is desirable to extend some of Fell's results, especially his main continuity theorem [4, Theorem 4.1], to the more general case. Such results are needed for the example in [2 ] in the study of the compactness of induced representations. We prove this extension in the present note. Throughout this paper, G will be a fixed second countable, locally compact group with left Haar measure ,i and corresponding modular function A, and A will be a fixed separable Banach *-algebra with a bounded, two-sided approximate identity. Let M(A) and cL(A) denote respectively the double centralizer algebra of A and the group of unitaries in M(A), each of these having the strict topology (see [1 ] for definitions), and let Auti(A) be the group of isometric *-automorphisms of A with the pointwise convergence topology. A twisting pair for (G, A) is a pair (T, ax) where a:GXG-*cUt(A) and T:G-Auti(A) are Borel functions such that: (1) T(x)a(y, z)a(x, yz)=a(x, y)a(xy, z) (the common value of these expressions will from now on be denoted fl(x, y, z)). (2) (T(x)T(y)a)a(x, y) =a(x, y)(T(xy)a); a(x, e) =a(e, y) =I. Received by the editors November 18, 1970. AMS 1969 subject classifications. Primary 4650, 4680; Secondary 2260.

It is proved in this paper that each sequence of measures with values in a topological group-defined on the Borel field of a finite or countable product of connected, locally connected, separable metric spaces-which is Cauchy convergent for all connected open sets is Cauchy convergent for all Borel sets, too. 1. Preliminaries. In this paper a topological group G is always assumed to be abelian. The system of neighborhoods of the zero element of G is denoted by 91(0). A sequence aneG, neN, is Cauchy convergent iff it is Cauchy convergent with respect to the uniformity {{(a,b)eG x G:a-be U}:Uel(0)}. Let X be a c-field on X; a function It: _-+G is a measure iff for all sequences of disjoint sets Aic-, icN, the sequence (C ,1 1u(Ai))nN converges to /AUjeN Ai). Let 0 be a subsystem of R, AeX, and ,u: 4-?G be a measure. We write 40nA={BnA:Bc-R} and 1t(RnA)={1t(B):BcRnAJcG. The set N:=fn {U: Uel(0)} is a closed subgroup of G. The quotient group GIN is a Hausdorff group and the natural homomorphism v: G GIN is continuous, open and vr-'G0==Go+N for each Goc G. Replacing G by GIN we may assume for our purpose without loss of generality that G is a Hausdorff group. If X is a topological space we denote by 7x the system of open szts in X and by tX the Borel field of X, i.e. the a-field generated by 'x. If Z=Xx Y and AcZ, i xA: ={xeX: (x, y)eA for some yeY} denotes the projection of A to X. A denotes the complement of a set A. 2. The main results. It is well known that convergence of (probability) measures for all connected open sets of the space R of real numbers does not imply convergence for all Borel sets (let e.g. Pn({n})= 1, neN). The following result shows, however, that this becomes true for all product Received by the editors September 30, 1971. AMS 1970 subject classifications. Primary 60B10; Secondary 28A35.

1. Helly's [8] theorem on intersections of convex sets ("If every k+1 members of a family of compact, convex subsets of Ek have a nonempty intersection, then the intersection of all the members of the family is not empty") has been generalized in various directions. Helly himself gave (in [9]) a generalization to families of not necessarily convex sets, in which the intersections of any 2, 3, * * *, k members are assumed to satisfy certain conditions (which are automatically fulfilled for families of convex sets). In other papers (e.g., [3; 7; 13 ]) problems related to Helly's theorem were considered under weaker assumptions on the number of sets with nonempty intersections, but restricted to families consisting of translates (or of homothetic images) of one convex set. Similar families have been considered also in connection with theorems of Helly's type for common transversals instead of common points (a list of references is given in [4]). On the other hand, it was shown [2] that if families of affine transforms of one set are considered, the convexity of the sets is necessary for the validity of Helly's theorem in its original form. The present paper results from an attempt to find whether there exists some theorem of Helly's type for nonconvex sets, in case only families consisting of affine or other appropriate transforms of one set are considered and, possibly, additional conditions imposed. But, as is easily seen by a slight modification of the example on p. 70 of [5] (or by Example 1 of the present paper), even if only families of translates are considered, and the sets assumed to consist of only two convex components, there exists no "critical number" corresponding to k+1 in Helly's theorem. Nevertheless, the following "individual" theorem is valid and, as shown by the examples in ?4, in many respects the best possible.

Introduction. Let K be an infinite perfect field of characteristic different from 2, let e3 be a quaternion division algebra over K, and let tt-denote the canonical involution of the first kind on Z. Let V be a finite-dimensional right vector space over Z. A quaternion skew-hermitian form H over Z is a sesquilinear form on VX V, i.e., H is a map from VX V to Z such that (i) H(x, Yl+Y2) =H(x, yi) +H(x, Y2) and H(x, yax) =H(x, y)a for all x, y, Y1, y2 in V and ai in ; (ii) H(x, y) = -H(y, x) for all x, y in V. Let {xl, . . *, x.) be a basis for V over Q. We say that H is nondegenerate if the reduced norm in Mn(Z) of the matrix (H(xi, xj)) is not zero. Associated to such a nondegenerate form H are 3 invariants, the dimension of V over 0, dime(V), the discriminant of H, b(H), and the Clifford algebra of H, G. Let G be a simply connected semisimple algebraic group (in some GL(m, X)) which is defined over K and let p:G-?GL(V/Z) be an absolutely irreducible representation of G defined over K into the group of all nonsingular Z-linear endomorphisms of V. We shall assume that there is a nondegenerate quaternion skew-hermitian form H on V which is invariant with respect to p (G). The purpose of this paper is to describe the Clifford algebra of the invariant form H in terms of p, G, and the Steinberg group associated to G. In a previous paper, we have described dimsz(V) and b(H) in such a way and have indicated how representations such as p arise [2, Theorem I.2 ]. The invariant -y(G) plays an important role in our study and so we recall some of its properties in ?1. In 2, we define the invariant Y using the representation p. Jacobson first constructed the Clifford algebra of a quaternion skew-hermitian form [4]. In this paper, however, we shall follow a method due to Satake [6]. We give some examples in ?3 with special emphasis on the case where G is absolutely simple.

This paper presents general conditions under which a subfamily may be selected from an infinite family of nonnegative, finitely additive measures such that this subfamily has the same cardinality as the original family, and such that the members of this subfamily are, in a certain sense, disjointly supported. The generalized continuum hypothesis is required for the general result, but not for a special case of this result which had previously been obtained by Rosenthal, and for which the present techniques yield a much shorter proof. This paper contains a short proof of a lemma about finitely additive measures which was first obtained by Rosenthal [2, Lemma 1.1, p. 16] in connection with a number of results on Banach spaces. Some illustration is then provided of the type of generalization which may be obtained with the present techniques. We shall understand an ordinal number to bea set and not an order type (see [1, ?4.3, p. 19]). As we shall rely heavily upon the axiom of choice, it will be sufficient for our purposes to define a cardinal number to be an initial ordinal number, i.e. an ordinal number which cannot be put into a oneto-one correspondence with any of its members. If F is any set, let F r I denote the cardinality of F, by which we mean the unique cardinal number which can be put into a one-to-one correspondence with F; let cf(F) denote the cofinality of I F 1, by which we mean the smallest cardinal number K such that I F I contains a cofinal subset of cardinality K; and let P(F) denote the power set of F, by which we mean the set of all subsets of F. To simplify notation we have trivially reworded Rosenthal's lemma, which now follows. 1. Lemma. Let F be an infinite set, let x e II be a family of nonnegative finitely additive measures defined on all subsets of F, and assume Received by the editors June 25, 1973. AMS(MOS) subject classifications (1970). Primary 28A10, 28-01, 04A20, 04-01; Secondary 46B99.

In this paper we prove that a Lie ring of characteristic prime to 2, 3 and 5, satisfying the fourth Engel condition, is nilpotent. 1. Let L be a Lie ring satisfying the fourth Engel condition, that is, for any a, x in L, ax4=0. Higgins [3] proved that if L has characteristic prime to 2, 3, 5, and 7, then L is nilpotent. Bachmuth et al. [1] showed that if L has characteristic 5, then L need not be nilpotent. The purpose of this note is to show that if L has characteristic prime to 2, 3 and 5, then L is nilpotent. 2. Let R be the (additive) endomorphism ring of L. Let D be the subset consisting of the inner derivations of L, X:a-+ax. D is a Lie ring under the product (X, Y) =XYYX. D is easily seen to be a homomorphic image of L under the map x-*X. THEOREM. A Lie ring L of characteristic 7, satisfying the fourth Engel condition is nilpotent. PROOF. For any X, Y in D, the following relations are evident from Higgins [3, Theorem 4]: (1) X3Y + 2XYX2 = 0. (2) X3Y 4X2YX + 6XYX2 4YX3 = 0. (3) X3Y3 -Y3X3 (4) YX3Y2 = 0. (5) Y2X3y = X3Y3. In the same paper Higgins discusses the process of linearization in his Lemma 1. Linearizing (1) we obtain, 18 (6) E Yli Y2aX Y33i y4o= 0. The ai's are some fixed permutations of {1, 2, 3, 4}, and Y1, Y2, Y3, Y4 are any elements of D. Received by the editors September 1, 1970. AMS 1969 subject classifications. Primary 1730.

In this paper it is shown that an upper semimodular poset without infinite chains satisfies the Jordan-Dedekind chain condition. This corrects an error in Theorem 14, p. 40 of [1] and generalizes that theorem. In [1, Theorem 14, p. 40], it is proved that the Jordan-Dedekind chain condition holds in any semimodular poset P of finite length. However, an error is made in the proof when the existence of a maximal chain y" connecting u to b is assumed. As can easily be shown by example, no such chain need exist. For instance, let P = { a, xi, ylu, b, d be the six element poset in which x1 and Yi cover a, u and b cover x1 and yl, and d covers u and b. If, however, P is a lattice, then the above cited proof is correct as -y" does exist. In this paper we give a correct proof to a generalization of the above theorem. We say that a poset P is upper semimodular if whenever distinct elements a, bC-P both cover an element cEEP then there exists dEP which covers both a and b. In [1 ] it is assumed that P has a first element 0 and is of finite length. We do not need these restrictions here. A dual definition can be made for lower semimodular posets and it is easily seen that dual arguments prove that our theorems hold for lower semimodular as well as the stated upper semimodular posets. Definitions of all remaining terms may be found in [1]. If a P we use the notation U(a) = { x (EP: x _ a }. LEMMA 1. Let P be an upper semimodular poset with -no infinite chains. If a CP, then U(a) has a largest element d. PROOF. Since P has no infinite chains, U(a) has a maximal element d. Suppose b 7 d is also a maximal element for U(a). Let A = [a, d]Cn [a, b]. Now A # 0 since aECA. Again since P has no infinite chains, A has a maximal element co. Now c0 a, and ci (1. Now a1 and bi both cover c0 Received by the editors May 18, 1970. AMS 1969 subject classifications. Primary 0620, 0630.

Introduction. Let wr be a finite projective plane of order n,-i.e., with n + 1 points per line. The author [2] has shown that if wr is doubly transitive and n is an odd nonsquare, then wr is Desarguesian. M. Hall and D. Hughes, in a paper to appear soon, have removed the restriction that n must be odd. In this paper, we show that if -T is dually transitive (see definition below), then it is doubly transitive.

It is shown, that if G is a LCA group and if H is a nondiscrete LCA group then there exists a proper closed subalgebra of the measure algebra of H (independent of the choice of G) in which the range of every homomorphism of the group algebra of G into the measure algebra of H is contained. Throughout this paper, G and H denote LCA groups and G and H denote their dual groups, respectively. X(H) is the set of all the locally compact group topologies of H which are at least as strong as the original one of H. For each % e X(H), if we denote by H7 a LCA group with underlying group H and topology t, the natural continuous isomorphism of HT onto H, x e ?Ti—>;t £ H, induces a natural norm-preserving im- bedding of L1(HT) into M(H), which we also denote by Lx(Hr). For the other notations and terminologies which we need in this paper, we follow (6). The author would like to express his thanks to the referee. His kind advice enabled the author to make this paper more readable. Theorem. If h is a homomorphism of Ll(G) into M(H), then there exist finitely many elements tx, t2, • ■ • , rn e X(H) such that the range of h is contained in 2?=i Lx(Hr'). For the proof of the theorem, we essentially use Cohen's results, which determine all the homomorphisms of L1(G) into M(H) by the notion of the coset ring and piecewise affine maps (cf. (1), (2), (3) and (6, Chapters 3 and 4)). If A is a homomorphism of L1(G) into M(H), Cohen's theorem asserts that there exist Y, an element of the coset ring of H, and a piecewise affine map a from Y into G such that

In this paper we determine (1) the class of all non- discrete LCA groups for which every proper closed subgroup is the kernel of a continuous character of the group, (2) the class of locally compact groups whose closed subgroups are totally ordered by in- clusion, and (3) the class of infinite LCA groups whose proper closed subgroups are topologically isomorphic. Since all these de- terminations involve only the most common LCA groups, we may regard our findings as characterizations of natural classes of these well-known groups. The program of deriving information about locally compact Abe- lian (LCA) groups from a knowledge of their closed subgroups has received attention in recent years; see, for example, references (3), (4), and (5). In this paper we shall state and prove three theorems characterizing some of the most common LCA groups by means of very natural hypotheses upon their closed subgroups. The LCA groups of which we shall make constant use are the circle T, the additive real numbers R, the integers Z, the cyclic groups Z{n), the quasicyclic groups Z(p°°), the £-adic integers JP and the p-adic numbers Fv. Precise definitions of all these groups may be found in (l); in particular, much detailed information on the groups Jp and Fp may be found in (l, §§10 and 25). Except where explicitly stated, all groups throughout are assumed to be LCA and Hausdorff topological groups. If the group Gi is topologically isomorphic to the group G2, we write Gi=G2. The character group of a group G is de- noted by G, and the kernel of a character 7 is written as ker y. Finally, if x is an element of a group G, we use the symbol (x) to denote the subgroup of G generated by x; the closure of this subgroup is written as (x). These findings constitute part of the author's doctoral disserta- tion submitted to Stanford University in 1969. The author is grateful to the National Science Foundation for financial support and to

As indicated by the title, the main result of this paper is a straightforward generalization of the following two theorems by J. Dieudonnen and by I. Amemiya and Y. Komura, respectively: (i) Every finite-codimensional subspace of a barrelled space is barrelled. (ii) Every countable-codimensional subspace of a metrizable barrelled space is barrelled. The result strengthens two theorems by G. K6the based on (i) and (ii), and provides examples of spaces satisfying the hypothesis of a theorem by S. Saxon. Introduction. N. Bourbaki [2] observed that if E is a separable, infinite-dimensional Banach space, then E contains a dense subspace M of countably infinite codimension which is a Baire space. R. E. Edwards [4] noted that since M is Baire, it is an example of a noncomplete normed space which is barrelled. Obviously, (i) and (ii) provide a plethora of such examples. It is apparently unknown whether every countable(or even finite-) codimensional subspace of an arbitrary Baire space is Baire; (for closed subspaces the results are affirmative). In the second paper [8], which follows, the authors give topological properties other than "barrelledness" which are inherited by subspaces having the algebraic property of countable-codimensionality. 1. The notation will be that used by J. Horvath [5]. If (E, F) is a dual pairing (E and F not necessarily separating points) then o(E, F) Presented to the Society, August 30, 1968 under the title On determining barrelled subspaces of barrelled spaces and November 9, 1968; received by the editors December 13, 1968 and, in revised form, June 10, 1970. AMS 1970 subject classifications. Primary 46AO7, 47A55; Secondary 46A30, 46A35, 46A40.

A commutative diagram is presented which relates the groups of concordance classes of diffeomorphisms f(S2n), F(CPU) and f(S2n+l). This diagram is applied to show that every equivariant diffeomorphism of S7 is concordant to the identity. It follows that the exotic 8-sphere, Y8, admits no smooth semifree SI-action with exactly two fixed points. Introduction. In this paper we shall present a commutative diagram (Theorem 1) involving the stable homotopy group 1711 and the groups r(Sn), r+(CPn) defined below. This diagram is then applied to show that every orientation preserving diffeomorphism of S7 which is equivariant with respect to the standard free action of S' on S7 iS concordant to the identity. Finally we show, using a result of R. Lee [3], that the exotic 8sphere, E8, does not admit a smooth action by S' which is semifree with exactly two fixed points. The paper concludes with a brief discussion of possible further applications of Theorem 1 to construction of smooth actions of S' on exotic spheres such that the actions are semifree with exactly two fixed points. The results of this paper are contained in the author's doctoral dissertation written at the University of Massachusetts. The author wishes to express his indebtedness to Professor J. C. Su for his generous help while the author was a graduate student. The author also acknowledges his gratitude for suggestions by a referee which resulted in a considerable shortening of the proof of Theorem 1. Preliminaries. Iffo and fi are diffeomorphisms of M onto N, M and N CO-manifolds, we sayfo is concordant tof1 if there is a diffeomorphism F:M x I->-N x I,I= [0, 1], suchthatforallxeMwehaveF(x, 0) = (fo(x), 0) and F(x, 1) = (f1(x), 1). The relation of concordance is an equivalence relation on the set of all diffeomorphisms of M onto N. If f :M -* N is a diffeomorphism its concordance class will be denoted [f]. If M is an orientable manifold there is a group, r(M), of all concordance Received by the editors March 15, 1971. AMS 1970 subject classifications. Primary 55E45, 57D60, 57E25.

Perhaps the two most basic phenomena discovered by the recent application of recursion theoretic methods to the developing theories of computational complexity have been Blum's speed-up phenomena, with its extension to operator speed-up by Meyer and Fischer, and the Borodin gap phenomena, with its extension to operator gaps by Constable. In this paper we present a proof of the operator gap theorem which is much simpler than Constable's proof. We also present an improved proof of the Blum speed-up theorem which has a straightforward generalization to obtain operator speed-ups. The proofs of this paper are new; the results are not. The proofs themselves are entirely elementary: we have eliminated all priority mechanisms and all but the most transparent appeals to the recursion theorem. Even these latter appeals can be eliminated in some "reasonable" complexity measures. Imnplicit in the proofs is what we believe to be a new method for viewing the construction of "complexity sequences." Unspecified notation follows Rogers [12]. 2iqi is any standard indexing of the partial recursive functions. N is the set of all nonnegative integers. 2iDi is a canonical indexing of all finite subsets of N: from i we can list Di and know when the listing is completed. Similarly, 2iFi is a canonical indexing of all finite functions defined (exactly) on some initial segment {0, 1, 2, , n}. ;,D, is any Blum measure of computational complexity or resource. Specifically, for all i, domain (Di= domain Xi, and the ternary relation (Pi(x)

The question of whether each self-universal crumpled cube is universal is answered negatively by presenting an example of a dogbone space which is not topologically E3 but which can be expressed as a sewing of two crumpled cubes, one of which is selfuniversal. C. D. Bass and R. J. Davernian [2] presented a brief paper indicating that the solid Alexander horned sphere is an example of a crumpled cube which is self-universal but not universal, thus answering negatively the question asked by C. E. Burgess and J. W. Cannon in [4] of whether each self-universal crumpled cube is universal. The validity of the example presented by Bass and Daverman depends on a claim that a certain upper semicontinuous decomposition of S3 into points and tame arcs, described in [2], is not topologically S3, which in turn depends on the validity of four lemmas which are stated in [2, §2]. The proofs of these four lemmas appear to entail nontrivial arguments which are not included in [2]. In this note, we present an example of a dogbone space, an upper semicontinuous decomposition of S3 into points and tame arcs whose nondegenerate elements can be expressed as the intersection of a tower of solid double tori, which is not topologically S3. The dogbone space can be described as the result of a sewing of two crumpled cubes, one of which is self-universal. Thus, the question asked by Burgess and Cannon in [4] is answered negatively. The argument will be based essentially upon work by Casier [5] and the author [1]. Some recent work by Eaton [6] includes a different proof that the solid Alexander horned sphere H, used in the example presented by Bass and Daverman, is not universal. This was done by sewing H to the crumpled cube F described by Stallings [7] so that the wild points of Bd //are sewn to the Cantor set of nonpiercing points of F. Other methods developed by Eaton, in papers cited in [6], should offer alternative ways Presented to the Society, October 30, 1971 ; received by the editors August 18, 1971. AMS 1970 subject classifications. Primary 57A10.

It is proved here that an odd number of the form paS6, where s is square-free, p is a prime which does not divide s, and p and a are both congruent to 1 modulo 4, cannot be perfect. A positive integer n is said to be perfect if (1) o(n) = 2n, where a(n) denotes the sum of the positive divisors of n. To date 24 perfect numbers have been discovered, all of which are even. Although no one knows whether or not any exist, many interesting results have been obtained concerning the structure of odd perfect numbers. The oldest of these goes back to Euler who showed that if n is an odd perfect number then p= p .'1 ... P2ft (2) n = p 2 pt wherep, pl, * * Pt are distinct odd primes and p=o 1 (mod 4). In 1937 Steuerwald [5] proved that not all of the pi in (2) can equal 1. Four years later Kanold [1] showed that the pi cannot all be equal to 2. In the same paper he also proved that the numbers 2#i +I (i=1, 2, * * *, t) cannot have as a common divisor any of the numbers 9, 15, 21 or 33. Recently McDaniel [3] has generalized these results by proving that 3 cannot be a common divisor of the 2,#i+ 1. The purpose of the present paper is to show that the pi in (2) cannot all be equal to 3. Thus, we shall prove the following result. THEOREM. If n =p"pl'p. p' is an odd number such that p x =_ 1 (mod 4), then n is not perfect. Our method of proof requires us to find the prime factors of some very large numbers. This part of the research was done using the CDC 6400 at the Temple University Computing Center. Received by the editors May 26, 1971. AMS 1970 subject classifications. Primary 10A20.

Let K[G] denote the group ring of G over the field K and let A denote the F.C. subgroup of G. In this paper we show that if K[G] satisfies a polynomial identity of degree n, then [G: Al] < n/2. Moreover this bound is best possible. If K[G] satisfies a polynomial identity of degree n, then it is known that [G: A] < 0o. In fact if K[G] is prime or if K has characteristic 0 then [G: A] < (n/2)2 by the results of [4]. In general we have [G:/A] < n! by the results of [1]. Thus the goal of this paper is to sharpen these to obtain the best possible bound, namely [G: A] < n/2. We follow the notation of [3]. 1. The abelian case. Throughout this section we assume that [G:/A] < cO and that A is abelian. Let xl = 1, X2, X3, . * Xm be a complete set of m = [G: A] coset representatives for A in G. LEMMA 1. 1. There exists a K-monomorphism p: K[G] -* K[A]m, where the latter is the ring of m x m matrices over K[A], satisfying (i) for a Ec A, p(a) = diag (all, ax2, * ax.), (ii) p(xj)ejj = eil, ellp(xT 1) = eii, where {etj} is the set of matrix units in K[A]m. PROOF. Since A is normal in G, {x11, x"2 1 1, x*-ml} is also a complete set of coset representatives for A in G. Set V = K[G]. Then clearly V is a left K[A]-module with free basis {x7', xi2, , x7m1}. Now V is also a right K[G]-module and as such it is faithful. Since right and left multiplication commute as operators on V, it follows that K[G] is a set of K[A]-linear transformations on an m-dimensional free K[A]-module V. Thus there exists a K-monomorphism p with p(K[G]) c K[A]m. Let a e A. Then x71a = (xT`axj)xT1 = axixT1; so clearly p(a) = diag (axl, a 2, . , axm). Now to compute ellp(xT1) we need only consider the first row of the matrix p(xT'). Since x1x-1 = x7-1 we see that this first row is precisely eli; so ellp(xi-1) = elieli = eli. Received by the editors February 18, 1971. AMS 1970 subject classifications. Primary 16A26; Secondary 16A38.

This paper shows that semitopological classes are subsemilattices of the lattice of topologies, and gives a new characterization for the finest topology in the semitopological class. Introduction. In [4] Levine defined a set, A, to be semiopen if there is some open set U so that Uc A c c(U), where c( ) denotes closure in the topological space. In [1] it was shown that if (X, i-) is a topological space, there is a finest topology [we shall call it F(r)] so that the semiopen sets are the same as for -r. If X is a set of points, let T(X) be the lattice of topologies on X. If zE T(X), let [v-] denote the equivalence class of all topologies which have the same semiopen sets as -r. [v-] is called a semitopological class of topologies on X. The object of this note is to show that if -r E T(X), [v-] is a subsemilattice of T(X) with respect to the usual join operation on topologies, and to give a new characterization for F(). 1. Semitopological classes are subsemilattices of the lattice of topologies. In [1] a set was defined to be semiclosed if its complement is semiopen, and semiclosure and semi-interior were defined in a manner analogous to the definitions of closure and interior. LEMMA 1.1. If (X, or) is a topological space, and if c( ) and i( ) denote the closure and interior, respectively, in (X, -r) while c*( ) and i*( ) denote the closure and interior in (X, F(r)), and sc( ) and si( ) denote the semiclosure and semi-interior in both, then if AccX, i*(c(A))c'sc(A). PROOF. If 0 e F(r) so that Oc: c(A), then consider 0 Cn(X-sc(A)). By Theorem 1.9 of [1], the intersection of an open set and a semiopen set is semiopen. Since sc(A) is semiclosed, (X-sc(A)) is semiopen; therefore O n (X-sc(A)) is semiopen in (X, F(-r)). Consequently, 0 n (X-sc(A)) = 0-sc(A) is semiopen in (X, i-). Now since Oc: c(A), we have 0 sc(A) c: c(A) sc(A) c: c(A) A. By Theorem 1.14 of [1], si(c(A)-A)= 0. Therefore, since 0-sc(A) is semiopen, 0-sc(A) = 0, so that O c sc(A). Received by the editors June 12, 1973. AMS (MOS) subject classifications (1970). Primary 54A05.

Let h h be an involution of a 3 3 -dimensional lens space L = L ( p , q ) L = L(p,q) . h h is called sense preserving if h h induces the identity of H 1 ( L ) {H_1}(L) . The purpose of this paper is to classify the orientation preserving PL involutions of L L which preserve sense and have nonempty fixed point sets for p p even. It follows that, up to PL equivalences, there are exactly three PL involutions on the projective 3 3 -space P 3 {P^3} , and exactly seven PL involutions on P 3 # P 3 {P^3}\# {P^3} .

If L \mathcal {L} is a countably productive normal base on a Tychonoff space X X , then η ( X , L ) \eta (X,\mathcal {L}) is an L ∗ {\mathcal {L}_ \ast } -realcompact extension of X X . R. A. Alo and H. L. Shapiro thus generalized the Hewitt realcompactification of X X . In the following paper, we extend this construction to T 1 {T_1} -spaces and show that it is an epireflection functor on an appropriate category. We are thus concerned with the question of the extendibility of a continuous map f : X → Y f:X \to Y to a continuous map g : η ( X , L X ) → η ( Y , L Y ) g:\eta (X,{\mathcal {L}_X}) \to \eta (Y,{\mathcal {L}_Y}) . We derive necessary and sufficient conditions therefor in the case when L Y {\mathcal {L}_Y} is a nest generated intersection ring on Y Y .

Although neither the result nor the e-transformation is new, a new order for the successive transformations is prescribed. From this follow some interesting properties which in turn imply the result. Let A, B and C be subsets of a finite abelian group G, with a, b and c the respective elements. Suppose further that A +B=C. DEFINITION. Let J ={x:x+Bc C}. We say A is full whenever A=A. Notice that A c'A and A +B=C. DEFINITION. Let AC={x:x+Cc C}. It is obvious that 0 AC, C+AC' C, AA c AC (in fact AA =AC if A is full). Also C+Sc C implies that SGAC. Hence AC+AC=AC, so that AC is a subgroup of G. We now use the usual Dyson e-transform. H. B. Mann's e-transform would serve equally well with only the obvious modifications in the proof, and indeed the proof was originally done using that transform. For e in A, a full set, and 0 E B another full set, A A' =A u (B + e), B B'= B (A -e), C C' = A' + B'. Also let A* = {a:a E A', a 0 A}, B*= {b:b eB, b r B'}. We have the following usual properties: PO. C, c C, P1. B*+e=A*, P2. IAI+IBI=IA'I+IB'I, P3. OeB'. Received by the editors January 13, 1972 and, in revised form, February 15, 1973. * This article has not been proofread by the author because the Amer. Math. Soc. was unable to locate him. The address given at the end of the paper is the last address given by the author. ? American Mathematical Society 1974

Let L be a diagonable subspace of an associative algebra A with identity over a field F; that is, L is spanned by a set of pairwise commuting elements, and the linear transformations ad x : a ↦ a x − x a x:a \mapsto ax - xa for x ∈ L x \in L are simultaneously diagonalizable. Denote the centralizer of L in A by C \mathcal {C} . A module V over A or C \mathcal {C} is L-weighted if for some nonzero v ∈ V v \in V and map λ : L → F , v ( x − λ ( x ) 1 ) n ( x ) = 0 \lambda :L \to F,v{(x - \lambda (x)1)^{n(x)}} = 0 for each x ∈ L x \in L , and x-weighted if for some nonzero v ∈ V , λ ∈ F v \in V,\lambda \in F and positive integer n, v ( x − λ 1 ) n = 0 v{(x - \lambda 1)^n} = 0 . In this paper we give conditions under which the following statements are equivalent: 1. All irreducible modules over A and C \mathcal {C} are L-weighted. 2. For each x ∈ L x \in L , some irreducible A-module is x-weighted and x is algebraic over F.

If ( X , T , π ) (X,T,\pi ) is a transformation group with locally compact phase group T T , there is a standard way to induce a transformation group on C ( X , Y ) C(X,Y) endowed with the compact-open topology, where Y Y is a uniform space. In this paper, we consider the case where Y Y is a topological group G G . The reverse construction under certain conditions is also considered.

where it is understood that/°(x)=x. Our main purpose here is to prove fixed-point theorems for nonexpansive mappings / for which the diameters of the sets 0(fn(x)) satisfy a condition introduced below, a condition which is suggested by a consideration of the Banach Contraction Principle. For such mappings /, compactness of M is seen to imply that every sequence of iterates {/"(x)} of x converges to a fixed-point of/ (which is not necessarily unique) while if M is a weakly compact, closed, and convex subset of a Banach space, then the existence of a fixed-point for/ is established. In the final section we show how the results of this paper lead in an indirect way to a generalization of Theorem 3 of [l].

In 1907, Hahn [2] showed that every (totally) ordered abelian group can be embedded in a lexicographically ordered, real function space. His proof occupies twenty-seven pages, not counting preliminaries, and may well be described as a transfinite marathon. For forty-five years, no one offered a simpler proof. But in 1952, Hausner and Wendel [3] gave a two-page proof of the same theorem for an ordered real vector space. In the present note, it is shown that the Hausner-Wendel proof applies equally well to the general case, with a few minor modifications, thus affording us an "accessible" proof of Hahn's fundamental theorem. Since this note is simply an appendix to [3 ], their definitions will not be repeated here. (Added in proof. Not until after this paper was printed did the author become aware of the fine work of Paul F. Conrad [4], who not only simplifies the proof, but extends Hahn's Theorem to partially ordered abelian groups and to even more general systems. Conrad bases his proof on the intrinsic notion of a "decomposition" of the given group G, instead of the extrinsic notion of an order isomorphism of G into an ordered function space. While basically equivalent, the two methods appear quite different, and perhaps it is just as well to have both in print.) We begin by reducing the general case to that of an ordered rational vector space. It was shown by Baer [1, p. 768], that any abelian group G having no element 5z? 0 of finite order can be embedded in a minimal "complete" abelian group V, i.e. one having the property that, for any v E V and any positive integer n, there exists x E V such that nx =v. Baer takes for V the set of all pairs (x, n), with xCG and n a positive integer, defining equality by

It is first proved that any uniformly continuous pseudo-ultrametric on a subspace of a non-Archimedean uniform space X has a uniformly continuous extension to X (which preserves total boundedness or separability). Then it is proved that every complete subspace of an ultrametrizable space X is a uniform retract of X. This has consequences concerning the extension of uniformly continuous functions. Isbell [6, Lemma 1.4] proved that every bounded uniformly continuous pseudometric p on a subspace of a uniform space X has an extension to a bounded uniformly continuous pseudometric on X. In this paper we consider the same problem when X is a non-Archimedean uniform space (see [7] for terminology) and p is a pseudoultrametric, i.e., p satisfies the strong triangle inequality p(x, y)

If S,, is a Sylow p-subgroup of the symmetric group of degree pn, then any group of order pn may be imbedded in Sn. We may express Sn as the complete product' C o C o ... o C of n cyclic groups of order p and the purpose of this paper is to show that any Sylow psubgroup of a classical group (see ?1) over the finite field GF(q) with q elements, where (q, p) = 1, is expressible as a direct product of basic subgroups En-C O C O ... o C (n factors), where Z is cyclic of order pr. (We assume always that p ;2.) Since C may be imbedded in S., we see that n is imbedded in Sn+r-l in a particularly simple way. The above r is defined by the equation q -1 =pt *where qI is the first power of q which is congruent to 1 mod p and * denotes some unspecified number prime to p. The case r = 1 is therefore of frequent occurrence, and then clearly SnSn Professor Philip Hall was my research supervisor in Cambridge (England) during the years 1949-1952 and it is a pleasure to acknowledge here my indebtedness to him for his generous encouragement.

Let h 1 {h_1} and h 2 {h_2} be PL involutions of connected, oriented, closed, irreducible 3-manifolds M 1 {M_1} and M 2 {M_2} , respectively. Let a i , i = 1 , 2 {a_i},i = 1,2 , be a fixed point of h i {h_i} such that near a i {a_i} the fixed point sets of h i {h_i} are of the same dimension. Then we obtain a PL involution h 1 # h 2 {h_1}\# {h_2} on M 1 # M 2 {M_1}\# {M_2} induced by h i {h_i} by taking the connected sum of M 1 {M_1} and M 2 {M_2} along neighborhoods of a i {a_i} . In this paper, we study the possibility for a PL involution h on M 1 # M 2 {M_1}\# {M_2} having a 2-dimensional fixed point set F 0 {F_0} to be of the form h 1 # h 2 {h_1}\# {h_2} , where M i {M_i} are lens spaces. It is shown that: (1) if F 0 {F_0} is orientable, then M 1 = − M 2 {M_1} = - {M_2} and h is the obvious involution, (2) if the fixed point set F contains a projective plane, then M 1 = M 2 = a {M_1} = {M_2} = {\text {a}} projective 3-space, and in this case, F is the disjoint union of two projective planes and h is unique up to PL equivalences, (3) if F contains a Klein bottle K, then F is the disjoint union of a Klein bottle and two points.

In a paper by J. Aczel [1], the following result is established: Let m(x1, X2, * * * , x,) be a continuous real function of the real variables xi (i = 1, 2, * , p, oo < a < xi < b < + oo) such that: (a) m(x, x, ... , x) =x for any x in (a, b), and (b) m(xl, x2, * * *, x,) is strictly increasing with respect to all xi. Then the sequence a, = m(an-P, an-P-1, , an-1) with the initial terms a1, a2, ap arbitrarily chosen in (a, b) is convergent. As corollaries of this theorem, the author proves the theorem of Enestrbm and Kakeya,' and some well known elementary results.2 In fact, the theorem of Enestrom and Kakeya is proved to be equivalent to the Aczel result if the function m is of the form

The object of this paper is to study the kernel of the map of the Brauer group of an integrally closed noetherian domain A into the direct product of the Brauer groups of the localizations of A at prime ideals. It is shown that this kernel is isomorphically contained in the torsion subgroup of the first cohomology group of the sheaf of Cartier divisors over Spec A. As a consequence, the author describes several new sets of conditions on A which guaran- tee that the kernel is trivial. We recall that M. Auslander and 0. Goldman (2) define the Brauer group of a commutative ring A as follows: let %(A) be the isomor- phism classes of central separable A -algebras, and SI0(^1) the subset of %(A) consisting of endomorphism rings of finitely generated projec- tive faithful ^4-modules. Let Si and 22 in 11(A) be equivalent if there are algebras 9ii and 9i2 in 3to(^4) such that Si ®A 9ii is isomorphic to ?2 ®a %. Then 93(^4), the set of equivalence classes of 21(^4) under this relation, is a group, the Brauer group of A. If S is a commutative yl-algebra, then there is a homomorphism from %$(A) to 33(5) reduced by the operation 2—>S ®a ?• In particular, for each prime ideal p of A there is a homomorphism from 33(^4) to $&(A»). Thus we have a homomorphism 0 from %5(A) to IlV33(ylp), where p ranges over the prime ideals of A, defined by mapping the class of 8 (which we will also denote 2 without fear of confusion), to {&>}. We would like to examine the kernel of this map in the case where A is an integrally closed noetherian domain. We are particularly interested in knowing when this kernel is trivial. In other words, we would like to answer the following ques- tion: given a central separable A -algebra L such that for every prime ideal p of A the localization Ln is isomorphic to Hoiru^F), F«) for some free finitely generated ^4p-module Fp, need there exist a finitely gener- ated projective A -module P such that L is isomorphic to Hom^(P, P)? According to results known at this time, it may be that the answer to

In a recent paper [6] Wermer considered the algebra C of all continuous complex valued functions on y, a simple closed anal-ytic curve bounding a region r, with ruy compact, on a Riemann surface F. He considered the subalgebra A of all functions in C which could be extended into r to be analytic on r and continuous on ruJy. Wermer showed that A was a maximal closed subalgebra of C which separated the points of y, and that the space of maximal ideals of A was homeomorphic to ruJy. In [2 ] Civin and Yood considered a class of subalgebras of complex commutative regular Banach algebras which become maximal closed subalgebras in the event the original algebra was the collection of continuous functions on a compact Hausdorff space. The object of this note is to demonstrate that such subalgebras possess a maximum modulus property possessed by A. To state the result obtained we recall certain definitions. The terms not herein defined may be found in [5]. Let B be a complex commutative regular Banach algebra with identity e and space of maximal ideals 9M(B). Let r: x-*x(M) be the Gelfand representation of B as a subalgebra of C((B)), the continuous function on 9M(B). We also denote irx by 2 and xrQ by Q for any subset Q of B. A subalgebra N of B is called determining [2] if irN is dense in irB, otherwise N is called nondetermining. A subalgebra of B is called a maximal nondetermining subalgebra if every larger subalgebra of B is deternining. A subset S of B is called a separating family on 2Z(B) if for each M1, M2 in 9YI(B), M1lM2, there exists an xES such that x(Mi) $x(M2). If P is an algebra of continuous complex valued functions vanishing at infinity on the locally compact space X, the smallest closed set (if it exists) on which each |ft with fGP assumes its maximum is called the gilov boundary of X with respect to P.

Here h(G), the Fitting height (also called the nilpotent length) of G, is as defined in [7]. 1I(G), the r-length of G, is defined in an obvious analogy to the definition of p-length in [2]. Higman [3] proved Theorem 1 in the case n =1 (subsequently, without making any assumptions on the solvability of G, Thompson [6] obtained the same result). Hoffman [4] and Shult [5] proved Theorem 1 provided that either p is not a Fermat prime or a Sylow 2-group of G is abelian. For p=2, Gorenstein and Herstein [1] obtained Theorem 1 if n< 2, and Hoffman and Shult both obtained Theorem 1 provided that a Sylow q-group of G is abelian for all Mersenne primes q which divide the order of G. Shult, who considers a more general situation of which Theorem 1 is a special case, recently extended his results to include all primes, but his bound on h(G) is not best-possible in the special case of Theorem 1. It also should be noted that Thompson [7] obtained a bound for h(G) under a much more general hypothesis than that considered in the other papers mentioned. Theorem 1 is a consequence of

In this paper a new representation is found for the root of unity occurring in the well-known transformation equation of the generating function for p ( n ) p(n) , the number of partitions of the positive integer n n .

This paper investigates the relationship between the p-length, l p ( G ) {l_p}(G) , of the finite p-solvable group G and the number, a p ( G ) {a_p}(G) , of orbits in which the subgroups of order p are permuted by the automorphism group of G. If p > 2 p > 2 and a p ( G ) ≦ 2 {a_p}(G) \leqq 2 , it is shown that l p ( G ) ≦ a p ( G ) {l_p}(G) \leqq {a_p}(G) . If p = 2 p = 2 and a 2 ( G ) = 1 {a_2}(G) = 1 , it is proved that either l p ( G ) ≦ a p ( G ) {l_p}(G) \leqq {a_p}(G) or G / O 2 ′ ( G ) G/{O_{2’}}(G) is a specific group of order 48.

Let f ( z ) = ∑ n = 1 ∞ a n z n f(z) = \sum \nolimits _{n = 1}^\infty {{a_n}{z^n}} , and set G ( z ) = f ( z − p ) − / 1 p = ∑ n = 0 ∞ g n p − 1 z 1 − n p G(z) = f{({z^{ - p}})^{ - /1p}} = \sum \nolimits _{n = 0}^\infty {{g_{np - 1}}{z^{1 - np}}} . This paper finds an explicit formula for g n p − 1 {g_{np - 1}} in terms of the a n {a_n} . Such a formula (apparently previously unknown) may be very useful in the theory of univalent functions.

We characterize by structure theorems the classes of all lattice-ordered groups in which (a) every prime subgroup is principal, (b) every proper prime subgroup is principal, and (c) every minimal prime subgroup is principal. These classes are also characterized by the structure of the root system of regular subgroups. In this paper we are concerned with the extent to which the structure of a lattice-ordered group (t-group) is determined by the structure of its prime subgroups. In [2], Paul Conrad has shown that each of the convex f-subgroups of an f-group G is principal if and only if G is a lex-sum of finitely many o-groups and each o-group used in the construction of G satisfies the ACC. In Theorem 1 we show that the same class of {-groups is characterized merely be requiring that all prime subgroups of G be principal. If we relax our requirements slightly by allowing G itself to be nonprincipal, then the class is slightly enlarged to include lex-extensions by certain o-groups of f-groups of the first class (Theorem 3). Finally, if only minimal prime subgroups are required to be principal, we get the class of all (-groups which are lex-extensions of finite lex-sums constructible from principal o-groups (Theorem 2). Furthermore, as in the original case, these classes can be identified by inspecting the lattice of convex f-subgroups. First we review here the basic definitions and facts about f-groups, all of which can be found in [2]. C is a convex [-subgroup of an 6-group G if C is a subgroup and sublattice of G and is convex in G (c?x?d and C, d E Cz>x E C). We denote by W(G) the lattice of all convex (-subgroups of G. A convex {-subgroup P of G is a prime subgroup of G (in brief, P is prime in G) if aAb=O implies that one of a and b is in P. The set of convex (-subgroups which include P always forms a chain, and every prime exceeds a minimal prime. P is a regular subgroup of G if it is a convex (-subgroup which is maximal with respect to not containing some Received by the editors September 25, 1972 and, in revised form, January 19, 1973. AMS (MOS) subject classifications (1970). Primary 06A55.

In this paper continuity of certain open monotone functions is obtained by assuming for the domain and/or range various combinations of the properties of a metric continuum, regular metric continuum, semilocal connectedness, and hereditary local connectedness. An open monotone connected function from a hereditarily locally connected separable metric continuum onto a separable metric continuum is continuous. If the domain is a regular separable metric continuum, an upper semicontinuous decomposition and resulting monotone-light factorization yield continuity of an open monotone function with closed point inverses. By a continuum is meant a compact connected space. A function f is monotone if point inverses are connected. If f is a function from X onto Y, the component decomposition X' of X induced byf is the collection of all components of sets of the form f 1(y), where y varies over Y. A function is connected if it takes connected sets onto connected sets. A continuum is regular provided every point has arbitrarily small open neighborhoods with finite boundaries [4]. It should be noted that this is not the same as a regular topological space as usually defined. THEOREM 1. If f is an open monotone connected function from the 1st countable space X onto the 1st countable semilocally connected space Y, then f is continuous. PROOF. If f is not continuous there exists an open set U in Y such thatf-'(U) is not open in X. Hence there is a point xCf1(U) and a sequence {xn } of distinct points in X -f'(U) such that xn--+x. Since Y is semilocally connected there exists an open set VC U such that YV has only a finite number of components. Since f(xn)E V for all n it follows that some component C of YV contains f(xn) for infinitely many n. By Theorem 2 of [I], ft1(C) is connected, and x is a limit point of f-1(C). Hence ft-(C)U{x} is connected but Received by the editors October 19, 1970. AMS 1969 subject classifications. Primary 5460; Secondary 5455.

I. Preliminaries. Throughout this paper po-group will mean partially ordered abelian group. A po-group G is semi-closed if geG and ng >0 for some n>0 implies g >0. G is directed if, whenever gi and g2 are elements of G, there is an element gCG such that g ?gi and g > g2. A subset B of G is lower directed (upper directed) if, whenever a, bCB, there is an element xCB such that x? a and x? b (x>a and x>b). B is a dual ideal of G if bCB and a>b implies aCB. If A is a convex subgroup of G, then a natural order is defined in G/A by setting XCGIA positive if X contains a positive element of G. All quotient structures will be ordered in this manner. For the po-group G, G+= {xCG: x>0}. A po-set S satisfies the Riesz Interpolation Property if, whenever Xl, * ..., xm, yi, * , y, are elements of S and xi?yj for 1_i