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C. Hobby and C. R. B. Wright [2] have just published the following Theorem A. However, their proof seems to contain an error.2 The notation of [2] is used except that G. is not reserved for the nth term of the lower central series of G: +/(G) denotes the Frattini subgroup of G; (G, H) means the group generated by the commutators g-lh-1gh where gEG, hEH; (A1, A2, * * *, A.+,) is defined inductively as ((A1, A2, . , A.), A.+,); HCG means that H is properly included in G.

In a recent paper (Square roots in locally euclidean groups, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 446-449), A. M. Gleason proved that, in a locally euclidean group G which has no small subgroups, there exist neighborhoods M and N of the neutral element e such that every element in M has a unique square root in N. The author clearly considered this result to be a step towards proving that, in such a group, some neighborhood of e is entirely filled by oneparameter subgroups of G. We shall establish here that he was justified in his expectation. As a matter of fact, our present proof requires only the local uniqueness of the square root, not its existence. This local uniqueness is established in the second part of the proof of Theorem 4 in Gleason's paper; and it may be observed that this part of the argument is independent of the assumption that the group be locally euclidean: it works equally well under the weaker assumption that the group be locally compact. We therefore have the following result

In the second paper in this series [2; 3], it was demonstrated that under suitable topological conditions on a space S (which are in particular always satisfied for a manifold), the problem of reconstructing S from its group G of homeomorphism onto itself may be reduced to the problem of reconstructing the points of S from G. If S can be reconstructed then every automorphism of G is inner, and it was shown in this way that every automorphism of G is in fact inner when S is the unit interval; this is the theorem of Fine and Schweigert [I]. The present paper is devoted principally to proving the FineSchweigert theorem for the disc. Given a space S topologically equivalent to the interior of the unit circle in the plane, the point set of S will be canonically reconstructed from the group G of homeomorphisms of S onto itself. A principal tool is the classical theorem that if g is an orientation-preserving homeomorphism of finite order of the disc S onto itself, then S may be mapped homeomorphically on the interior of the unit circle in such a way that g becomes a rotation, [4]. Since a homeomorphism of odd order is necessarily orientation-preserving, it follows that an element of odd order (different from one) of the group G of homeomorphisms of S onto itself has precisely one fixed point, which will be called its center. If x is any point of S, let x' denote the set of all elements of G of finite odd order (different from one) having x for center. The collection S' of all sets x' is in canonical one-one correspondence with the point set of S. Our problem will be at an end if S' can be exhibited as a structure derived from the purely algebraic structure G. To this end it is sufficient to be able to decide within the structure of G when two elements of odd order (different from one) have distinct centers. This occupies the remainder of the paper. Henceforth, when an element of G is said to be of odd order, it will be understood that the order is different from one.

In [2], I exhibited a set of (three) generators for the collineation group, G2, of the free plane T2. In the last section of that paper, I claimed to have a set of relations for the group, but had omitted them due to the length of the proof that they sufficed to determine the group. Several requests for the relations (and my own realization that this particular group could not easily be studied by the purely geometric presentation of [2]) led to the preparation of the present paper, in which the proof of the sufficiency of the relations is somewhat different from the original proof. I would like to thank Professor W. H. Mills for several helpful discussions concerning the present proof. This paper was written as an extension of [21, and the notation and terminology are exactly the same as in [2 ] except in one instance, and the difference is carefully explained when the subject is introduced. In studying the group G2, it seems necessary to make heavy use of the geometric properties of ir2, so we begin with an examination of certain relations between the generating configurations in 7r2. As in [2 ], we have wr0-a set of four points, no three of which are collinearwhich generates 7r2, and relative to which we assign a stage to every point and line of 7r2. A generating configuration, C (consisting of four points of 7r2 no three collinear, which generate 7r2), is said to be of stage k if each of the four points of C has stage < k, and if some point of C has stage k. We have the following result.

The question of when transitive translation functions exist for fiber bundles was raised by W. Hurewicz. The answer this paper supplies is the following. If a bundle over a finite polyhedron has a structural group G with no small subgroups then it has a transitive translation function if and only if it is equivalent in G to an H bundle where H is a totally disconnected subgroup of G. The central result of this paper is that if T is a transitive translation function and the structural group has no small subgroups, then r(w) depends only on the homotopy class of co. All spaces we consider will be Hausdorff spaces; path spaces will have the compact-open topology. REMARKS. For path spaces one may take as a basis all sets of the form Nf-nl1, ([(i-1)/2 , i/28], U). A sequence co. converges to the constant x0 in the path space XI if and only if every neighborhood U of x0 in X contains all but a finite number of the sets cn(I). That this is not true for Moore paths pre

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.

Furthermore, those of level $p are not congruence groups. It is well known that r contains infinitely many normal subgroups of finite index which are not congruence groups; see for example papers [3], [4]. However, all of these groups have the common feature that they are "lattice subgroups" (in Rankin's terminology) of some normal congruence group, and so are not maximal. The results of this paper imply that r contains infinitely many maximal normal subgroups of finite index which are not congruence groups, a somewhat surprising fact. The question which originally motivated this paper was the following: Which of the known simple groups have a representation as a modular quotient group, or equivalently, may be generated by two elements, one of period 2, the other of period 3? Call such a group a r-group. Then the groups LF(2, p) are certainly r-groups. However, it is not known for example when the alternating group is a r-group. In his lecture notes on Fuchsian groups [2], Macbeath poses a similar question for H-groups, and makes some remarks about the linear fractional groups which in fact form the basis of this paper, and which we are happy to acknowledge here.

If a compact Lie group acts effectively on a manifold, then local cross-sections to the orbit structure can be constructed. They are called slices. We prove that if a compact group acts effectively on a manifold, but it is not a Lie group, then a slice cannot exist at every point. Introduction. Recent papers of R. F. Williams have focused new attention on the Hilbert-Smith conjecture: If a compact group G acts effectively on a manifold, then G is a Lie group. Much interest has centered around properties which arise if one assumes that the conjecture is false, and that the p-adic group A, acts effectively on a manifold M. The purpose of this paper is to show that if such an action occurs, then for any reasonable definition of a slice, there is at least one point on the manifold at which a slice fails to exist. DEFINITION. Let (X, G) be a compact G-space, and let xeX. A slice at x is a subset Y of X containing x and satisfying: (i) Whenever g E G and y E Y, g(y) E Y?g E Gx. (ii) If Q is an open neighborhood of eCG/G. which serves as a local cross-section to the cosets of G. at the identity of G, then (g, y)-?g(y) defines a homeomorphism of QX Yonto Q(Y). Moreover, Q(Y) is open in X. REMARK. Equivalent definitions of a slice for the case where G is a compact Lie group, may be found in Montgomery and Yang [4], and in Chu [1], and Palais [6], for noncompact Lie groups. DEFINITION. Let Gk denote a cyclic group of order pk, and let gkA; GA be a fixed generator. For each pair of integers i, j>: 0 with i?j we let cij::Gi -Gj be the homomorphism which arises by sending gi-?gj. Note that cii is the identity map on Gi. With the positive integers as directed indexing set, it is easily verified that the collection of groups and homomorphisms { Gi, cij } forms an inverse system. The group A, = lim, { Gi } is called the p-adic group. Its topology and group operation are those which are induced from the product X5Gi, and from the standard theorems on inverse limits we find that Ap,is abelian, Hausdorff, and compact. Received by the editors March 5, 1970. AMS 1969 subject dassifications. Primary 2240, 5480.

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

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.

Projectors of finite, solvable groups were defined by Schunk [6]. In this paper, the author defines generalized projectors of finite groups. They are the same as projectors in the case of solvable groups. The existence of this class of subgroups is shown for i7T-solvable groups satisfying certain properties. Let X be a class of groups, which is closed under isomorphisms and 1 & X. DEFINITION. S is x-maximal in G if (i) S is a subgroup of G, (ii) S E X (iii) S(-TszG & Tec XS=T. DEFINITION. S is a X-projector of G if (i) S is a subgroup of G, (ii) H< G-?HS/H is X-max in G/H. DEFINITION. X is called a Schunk class if X satisfies the following conditions: (a) if G E X, H< G then G/H E X, (b) whenever all primitive factor groups of a group G are in X then so is G in X. Schunk [6] proved the following theorem. THEOREM. If X is a Schunk class then every finite, solvable group G has a X-projector and all X-projectors are conjugate. DEFINITION. Let S be a class of subgroups of a finite group G. Then we say S is a characteristic class of conjugate subgroups if (i) H E S and a an automorphism of G then Ha E S, (ii) Ha is a conjugate of H under an element of G i.e., 3g E G such that HHa-H . DEFINITION. A finite group G is called -r,-closed if it contains a normal subgroup N which is a 7T-group and GIN is a 7T'-group. All groups considered in this paper are finite. Let X be any Schunk class consisting of solvable, -r,-closed groups. In this paper we will show that if the 7T'-subgroups of a 7T-solvable group G have X-projectors and they form a characteristic class of conjugate subgroups then G has X-projectors and the X-projectors form a characteristic class of conjugate subgroups. We cannot expect that all X-projectors are conjugate as is clear from the case when G is a finite, nonabelian simple Received by the editors October 18, 1971 and, in revised form, April 26, 1972. AMS (MOS) subject classifications (1969). Primary 2054. ? American Mathematical Society 1973

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.

Introduction. In this paper we will consider the structure of a nonsolvable group satisfying the condition which I will denote by IC. DEFINITION. A finite group G is said to be an IC group if for any two nonincident subgroups H and K, HC'K has only cyclic Sylow groups. REMARK. Two subgroups H and K are nonincident if H?7K and K?(H. It follows easily that both subgroups and factor groups of IC groups are themselves IC groups. The result we will prove is the following.

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.

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 previous papers [4; 5; 6] we constructed and studied many classes of operations on groups which have the following properties: (1) Associative; (2) Commutative; (3) The resultant group contains isomorphic images of the original factors and is generated by them; (4) In the resultant group the intersection of a factor with normal subgroup generated by the remaining factors is the unit element. Golovin [ 3 ] showed how to obtain all operations satisfying the last three conditions and called such operations regular multiplications. Below we construct for every property (d, applicable to groups, a corresponding regular multiplication. If (d persists homomorphically and ceases freely, we shall show that this multiplication is associative. (d is said to persist homomorphically if from the fact that a group satisfies (P it follows that every homomorphic image of the group also satisfies (Y. On the other hand, (P is said to cease freely if for every set of groups satisfying (P their free product does not satisfy (P. We introduce a number of preliminary concepts which are all contained in Moran [6]. N(X) is said to be a normal subgroup function if it associates with every group G a normal subgroup N(G) of G. However, we are interested in the following more restrictive class of normal subgroup functions. N(X) is said to be a strongly characteristic subgroup function if N(X)o ? N(Xq) for every homomorphism 4 of X. This is equivalent to the condition that the normal subgroup be prescribed in such a manner that when extra defining relations are imposed on the group then the elements which originally belonged to the subgroup continue to remain therein. It seems that most of the well known normal subgroup functions, such as the Frattini subgroup, are of this type. Let F denote the free product of an arbitrary set of groups Ga, aECM, and N(X) be a normal subgroup function, then corresponding to N(X) we have the normal subgroup [N(Ga), G# ]F of F. [N(Ga), G# ]F is the normal subgroup of F generated by all commutators of the form [ga, g#], where gaCN(Ga), gEzG# and a and d are distinct elements of the index set M. We now define the N*-muitiplication of the groups Ga, ca EM, byl

Let al, , a. be free generators of the free group Fn, cl, cn those of the free group Fn, N an isomorphism of Fn onto Fn,t, and bi = N(a,) the maps of the a, under N. Then the bi are generators not only of Fn' but of any quotient group G'. I shall say that the generators bi of G' arise from the a, by a Nielsen transformation, N. In case Fn = Fn,tX N is (Nielsen) automorphism of Fn but need not be an automorphism of G = G'; nor is every automorphism of G expressible in terms of some N acting on a given set of generators. However, a given automorphism of G can be written as a Nielsen transformation acting on a suitably large set of generators (Theorem 2). I shall discuss this and related matters in the present paper. The following definitions and notation will be used. G= I{a,, * * *, an; R1, * * *, Rk} = {a; R} will denote the group given on n generators a, and k generating relations

A representable /-group G can be embedded into a projectable (strongly projectable) /-group; an essential extension of G which is minimal with respect to being projectable (strongly projectable) is unique. In this paper these projectable and strongly projectable "hulls" of G are represented using direct limits. If G is an /-ring, or /-ring without nonzero nilpotent elements, then so are these hulls. 1. Introduction. An /-group G is said to be representable if it can be embedded into a cardinal product of totally ordered groups, projectable if each polar subgroup of the form g" for g e G is a cardinal summand of G, and strongly projectable if every polar subgroup of G is a cardinal summand. If G is projectable then each polar subgroup of G is a normal subgroup and so G, and hence each /-subgroup of G, is representable (4, Theorem 1.8). Conversely, if G is representable then G is a dense /- subgroup of its orthocompletion (2, pp. 116, 125) which is strongly projectable. If G is a large /-subgroup of a strongly projectable (projectable) /-group K then the intersection H of all strongly projectable (projectable) /-sub- groups of A'containing G is strongly projectable (projectable), and Conrad (5) has shown that H is the unique (up to /-isomorphism extending the identity map of G) minimal strongly projectable (projectable) essential extension of G, and called H the sp-hull ip-hull) of G. In this note we wish to give direct limit representations of these hulls in a fashion similar to that in which Conrad (3, pp. 455-457) has represented the orthocompletion

Let G be a (totally) ordered (abelian) group. A Gmetric space (X, p) consists of a nonempty set X and a G-metric p: Xx X-G (satisfying the usual axioms of a metric, with G replacing the ordered group of real numbers). That the amalgamation property holds for the class of all metric spaces is attributed, by Morley and Vaught, to Sierpifiski. The following theorem is proved. THEOREM. The class of all G-metric spaces has the amalgamation property if, and only if, G is either the ordered group of the integers or the ordered group of the reals. The purpose of this paper is to prove that the amalgamation property fails for the class of all G-metric spaces, for every (totally) ordered (abelian) group G which is not equal to the additive group of the integers or that of the real numbers. The amalgamation property, in its abstract form, was first formulated by Fraisse [3] in connection with embedding problems. It has been studied by Jonsson [6], [7], [8] and [9], and Morley and Vaught [12] in connection with the general theory of homogeneous-universal structures in Jonsson classes. Not too many examples of classes of relational systems are known for which the amalgamation property fails. Among them the following are included: (a) the class of semigroups and hence of rings (Kimura [11], J6nsson [6], [7], Howie [5]); (b) the class of modular lattices (Jonsson [10]); (c) the class of i-groups (Pierce [13]). To these, because of our theorem, we can now add a whole family of such examples, among which is the class of all metric spaces on which the metric takes on only rational values. Classes of metric spaces were studied by Urysohn [17], [18] and Sierpin'ski [14], [15] from the point of view of the existence of universal Presented to the Society, July 6, 1971; received by the editors April 25, 1972. AMS (MOS) subject classifications (1970). Primary 06A55, 08A05, 54E35.

Theorems of Gaschiutz and Baer concerning the Frattini subgroup, the hypercenter, and nilpotent and supersolvable subgroups are extended to normally closed saturated formations. The hypercenter, derived group, and Frattini subgroup are all related to nilpotency of a finite group. In [4], Gaschiitz studied their relations, and found a sufficient condition for a normal subgroup of a group to be nilpotent. By replacing the notion of centrality with a-centrality, we give theorems which extend the results of Gaschuitz to normally closed saturated formations containing all nilpotent groups, and strengthen his nilpotency criterion. A theorem of Baer, on supersolvably immersed subgroups, is also extended to saturated formations. In this paper, all groups are finite. A class of groups is understood to include, with G, all groups isomorphic to G. If M is a subgroup of G, core(M) is the intersection of the G-conjugates to M, and is the largest normal subgroup of G contained in M. A chieffactor of G is a quotient H/K, where K< G and H/K is minimal normal in GIK. If H/K is a chief factor with L

In a group G, we let S(x) denote the normal subsemigroup generated by x and S'(x) = S(x)YJe. Following Fuchs [2], an element x of G is said to be generalized periodic (GP) iff eES(x) and nongeneralized periodic (NGP) otherwise. A group is said to be an 0* group iff every partial order on G can be extended to a full order. Ohnishi [9] considered the following group-theoretic condition: (i) For all x, y, t in G, if xES(t) and yES(t), then S(x)f\S(y) $0. He proved that a group G is an O* group iff every nontrivial element of G is NGP and G satisfies condition (i). In this paper we will study groups which satisfy condition (i) and will also consider two other conditions, one of which is equivalent to (i) and another which is implied by (i). We remark first that if G satisfies (i), it is known and easily verified that any homomorphic image of G also satisfies (i). Thus a homomorphic image of an 0* group is an 0* group iff every nontrivial element is NGP.

NOTATION AND DEFINITIONS. A group G has property E if it can be embedded in a sfield D and property EE if every automorphism of the group G can be extended to be an automorphsim of D. For a more complete discussion see [5 ]. The result on locally finite groups depends mainly on a result of Amitsur (see [I]). A complete discussion of the following notation can be found in his paper. 7r will denote the set of all primes and ri the set of all odd primes p such that 2 has odd order mod p. Let m and r be relatively prime integers. Put s = (r -1, m), t = m/s and n = minimal integer satisfying rn= 1 mod m. Denote by Gm,r a group generated by two elements A and B satisfying Am=1, Bn=At and BAB-'= A. Denote by G., a group G which has a countable ascending tower of subgroups Hi: O < i< oo } such that G = U1 Hi and each Hi is isomorphic to Gmi,ri. T*, 0*, 1* will denote the binary tetrahedral, octahedral and icosahedral groups. Let p be a fixed prime dividing m. a= aP is the highest power of p dividing m. 7, is the minimal integer satisfying rap= 1 mod (Mp-a). AuP is the minimal integer satisfying r/,P=p'2 mod (Mp-a) for some integer ,u. bp is the minimal integer such that ppr_ 1 mod (mp-a).

In this paper we generalize to groups of Galois type some results of P. Hall on finite solvable groups [l; 2; 3]. We need, in a modified form, some results of van Dantzig: the definition of supernatural numbers (which are related to van Dantzig's universal numbers) and Theorem 5, which he proved for ordinary ?>-Sylow subgroups [6]. Lemmas 1 and 4 and the method of proof in Theorem 5 are due to Tate [S]. A topological group G is of Galois type if it is compact and totally disconnected. In any Galois type group the open normal subgroups form a neighborhood base at the identity. Every closed subgroup is the intersection of the open subgroups containing it [4]. Whenever M and N are open normal subgroups of G and NZ)M we shall write 0jv for the natural homomorphism of G/M onto G/N (these quotient groups are finite) and

Throughout this discussion S will denote a compact connected topological semnigroup and E will denote the set of idempotents of S. The problem to be considered concerns a question posed by Professor A. D. Wallace. In [I], Wallace proves that if S has a left unit, if I is a closed ideal of S, and if L= D2 or if L is a closed left ideal of S, then Hn(S)_Hn(IQ_UL) for all integers n, where Hn(A) denotes the nth Alexander-Cech cohomology group of A with coefficients in an arbitrary but fixed group G. If S is assumed to have both a left zero and a left unit, then it follows that each closed left ideal L, of S is acyclic; that is, HP(L)= 0 for all p 1. A dual statement holds for closed right ideals if S has a right unit and right zero. A generalization of the case in which S has a left, right, or two-sided unit, is to require that S=ES, S=SE, or S=ESE, respectively, and Wallace has aslced: "If S has a zero, are closed right or left ideals of S necessarily acyclic in the more general situation?" [3]. A negative answer to this question is given here by way of examples, and a theorem is proved giviilg a necessary and sufficient condition for closed right ideals of S to be acyclic, assuming S=ESE and S has a zero. Following the proof of this theorem is an example of a semigroup not satisfying this condition. The above-mentioned example shows that even though S is acyclic, it is not necessarily true that all closed right ideals of S are acyclic. Trhus the question remains as to whether S is acyclic if S= ESE and S has a zero [3 ]. Wallace proves in [2 ] that for such a semigroup S, HI(S) = 0, however, an excample is given here of a semigroup S with zero, S=ESE and H2(S) ?G for all groups G, showing that this question also has a negative answer. Two further examples are included in this paper which show what can occur if one only assumes that S= SE, or S= ES. One example is of a semigroup S with zero, S= SE and HI(S)>--G for all groups G and the other is an example of a semigroup S with zero and left unit and S contains a closed right ideal R with HI(R)-G. DEFINITION. Let T be a semigroup, aC T and R(a) the closed right ideal of T generated by a. Then a is said to be right codependent on

A regular (weakly) transitive translation function for a fibre bundle induces an H-space antihomomorphism from the loop space of the base into the group of the bundle. The question when such translation functions exist has had three answers to date. E. Brown [1, p. 226] states that every bundle with paracompact base admits weakly transitive translation functions. However, E. Fadell has pointed out an essential error in the proof. (Namely, [1, p. 227] v.(a), v.(f8) and v,,(a+o) are unrelated.) In Steenrod's book [4, p. 59] we find that if a bundle has totally disconnected group, then it admits transitive translation functions. J. Schlesinger [3] has proved a converse to this result for a restricted class of bundles. Schlesinger's theorem fails to contradict Brown's claim for the following reasons: (1) Brown employs Moore paths while Schlesinger uses ordinary (unit domain) paths. (2) Schlesinger discusses transitivity while Brown claims only weak transitivity. This paper removes the first of these distinctions; namely we prove

1. Let G be a connected locally compact group and let G' denote the closure of the commutator subgroup of G. G' is called the derived subgroup of G. Consider the derived series of G, that is, the sequence of subgroups Go, G1, , G ... where Go=G and Gn+?=G'. Each Gn is connected and this sequence becomes stationary at some finite stage,2 that is, for some n, G, =G,+ We define G, to be the "core" of the group. The purpose of this paper is to prove the following

1. Let (X, T, 7r) be a topological transformation group, where X is a nontrivial Hausdorff space and T is a topological group which leaves an end point e of X fixed. In [3], Wallace proved that if T is cyclic and X is a locally connected continuum, T has a fixed point other than e. Then in [4], Wallace asked the following question: If X is a peano continuum and T is compact or abelian, then does T have a fixed point other than e? In [5] Wang showed that if T is compact, and X is arcwise connected, then T has a fixed point other than e. Then Chu [1 ] showed that T has infinitely many fixed points. Chu began the study of the abelian case in [2]. In this paper we show by example that X may be a peano continuum and T may be a countably generated abelian group which has e for its only fixed point, ?2. Thus in general the answer to Wallace's question in the abelian case is no. However, if T is a generative group, and X is compact and arcwise connected, then T has a fixed point other than e, ?3. We also show by example that T may be a finitely generated nonabelian group which has e for its only fixed point, ?4.

In this paper the Hughes conjecture for finite pgroups is proved in the case of groups which are nilpotent of class 2p-2. If G is a group and p is a prime number then the Hughes subgroup HpiG) of G is by definition the subgroup generated by all elements of G that do not have order p. The conjecture of Hughes was that if G>HpiG)>l then HviG) must have index p in G; see [o]. Hughes showed that the conjecture is true for p = 2 and all G, and Straus and Szekeres [lO] proved it for p = 3 and all G. Hughes and Thompson [7] proved the corresponding result for all p and for G finite and nonnilpotent. The most interesting cases which remain are those in which G is a finite 7^-group with p ^ 5 ; throughout this note we shall take G to be a finite £-group, and our results will be of most significance when pit5. We may remark at this point that it is a nontrival exercise to construct a £-group GP for each prime p such that Gp>HPiGp) > 1. Though Wall [ll] has shown by means of an example that the Hughes conjecture is false for finite 5-groups and p — 5,it is nevertheless of some interest to find sufficient conditions for the conjecture to hold. Thus Zappa [12] has shown that all is well when G is nilpotent of class p. The main result of the present note is a substantial strengthening of this: Theorem. The Hughes conjecture is valid for finite p-groups of nilpotency class 2p — 2. It seems likely to us that class 2p — 2 cannot be replaced by a much greater class without falsifying the theorem. The proof of this theorem depends heavily on the main result of [8], which we state again in the interests of clarity: Theorem [8]. If \G:HPiG)\^p2, V„(G)£t(G) and y2p(G) = l, thenVPiHpiG)r\y2iG)) = l. Received by the editors February 3, 1970. AMS 1970 subject classifications. Primary 20D15, 20D25.

A well-known sufficient condition that a group of homeomorphisms, H H , from a topological space X X onto itself be a topological group relative to the topology of pointwise convergence is that X X be uniformizable and H H be equicontinuous. In this paper we prove an analogous condition in which the space X X is assumed to be only regular instead of completely regular (uniformizable). This is accomplished by means of the concepts of semiuniformity and semiequicontinuity introduced here.

A permanent group is a group of nonsingular matrices on which the permanent function is multiplicative. We consider only permanent groups which contain the group of nonsingular diagonal matrices. If the underlying field is infinite of characteristic zero or greater than n, then each such permanent group consists only of matrices in which exactly one diagonal has all nonzero entries. A permanent group is a group of nonsingular matrices on which the permanent function is multiplicative. One example is 3>„(F), the group of nonsingular n X« diagonal matrices over the field F. Marcus and Mine [3] conjectured that A„, the groups of «xn matrices of the form PD where P is a permutation matrix and D e 9n(F) is a maximal permanent group. In this conjecture the field F was not specified and the first author [1] verified the conjecture for the field of complex numbers. In this paper we consider the set 3^n(F) of those permanent groups which contain @n(F), and characterize ^n(F) when Fis an infinite field with char F—0 or >«. In [2] we defined the sets/n(F) of all« x« matrix groups G of nonsingular matrices over F satisfying: (i) 2?>n(F) „(F) where "»" denotes the Hadamard product. Here, we are also concerned with the set ¥i n(F) of «x« nonsingular matrix groups over F such that G = H ■ K={hk, h e H,he K} where: (i) H es/„(F), (ii) A" is a group of «xn permutation matrices, and (iii) (PHP-l\PeK)esfn(F). Note that 2JJF)^H for every Hes/n(F) so that 2n(F)^G for every G e

Introduction. Let G be a locally compact, connected topological group (satisfying the second countability axiom). Let G* be a compact space which contains a dense subset G' homeomorphic to the space G and is such that G*-G' is totally disconnected. Then, Freudenthal has proved [1, Satz 1X, p. 277]1 that the set G*-G' consists of at most two distinct points. Actually, Freudenthal's theorem even for topological groups is more general than here stated, and this theorem is an application to group spaces of a wider theory of "ends" of topological spaces. However, we shall quote only so much of Freudenthal's results as are necessary to this paper. It will be convenient to regard G' as identical with G so that G is topologically imbedded in G*. We shall call a locally compact, connected group G two-ended if a G* exists such that G* G consists of two distinct points. The simplest example of such a group is the additive group of reals. Other examples are afforded by the direct product of this group and any compact connected topological group; it is likely that these are the only examples. The principal objective of this note is the following theorem.

Let G be a group and H p ( G ) Hp(G) the subgroup generated by the elements of G of order different from p. Hughes conjectured that if G > H p ( G ) > 1 G > Hp(G) > 1 , then | G : H p ( G ) | = p |G:Hp(G)| = p . In this paper it is shown that if G is a finite p-group and certain central factors of G are cyclic or if the normal subgroups of G of a certain order are two generated, then the Hughes conjecture is true for G.

Let F be a subfield of an algebraically closed field A of characteristic 0, S a finite subset of A disjoint from F, K a subfield of A containing F and maximal with respect to disjointness from S, L a finite extension of K, and G=G(L/K) the group of automorphisms of L/K. Quigley [4] and McCarthy [1] obtained precise information about G in the case where S has one or two elements, respectively (they handled the characteristic p case also). Theorem 1 of this paper shows that there is some restriction on G in the general case. In particular (Theorem 2), G is solvable if S has at most twenty elements.

The purpose of this paper is to study a class of finite groups whose subgroups of prime power order are all pro-normal. Following P. Hall, we say that a subgroup H of a group G is pro-normal in G if and only if, for all x in G, H and Hz are conjugate in (H, HX), the subgroup generated by H and HX. We shall determine the structure of all finite groups whose subgroups of prime power order are pro-normal. It turns out that these groups are in fact soluble t-groups. A t-group is a group G whose subnormal subgroups are all normal in G. The structure of finite soluble t-groups has been determined by Gaschuitz [1]. Are soluble t-groups precisely those, groups whose subgroups of prime power order are all pro-normal? The answer is yes. Thus our study furnishes another characterization of soluble t-groups. Except for the definitions given above, our terminology and notation are standard (see, for example, M. Hall [2 ]). We first prove two general facts about pro-normal subgroups.

Certain aspects of Mackey's theory of virtual groups were fitted into a cohomology theory for ergodic groupoids in a previous paper by the author. Here we relate the groupoid cohomology for ergodic groupoids that arise (by Mackey's construction) from ergodic actions of countable groups to the usual group cohomology. In case the countable group is a free group, we find that the groupoid cohomology in dimension > 1 is = { O }I. Given an ergodic action of a countable group, G, on a standard Borel space, X, with G-invariant measure class, C, we form the ergodic groupoid ff as in [4, Example 3.2b] (cf. [2]), the transitive ergodic groupoid XXGXX X=-, and the natural monomorphism, h: 5--f (inclusion map). The group cohomology referred to below is as developed in [3]. Given an abelian Borel group A (as in [4]), we define Al= { Borel functions, f:X-*A } mod equality a.e., with the abelian group structure defined by pointwise addition of functions. The action of G on Al is given by (f) =f y-1. With the trivial action of G on A, the map A->A,; a-->constant function--a, is a G-module homomorphism, and we obtain the short exact sequence of G-modules 0 -A -> Al --> A/A ->0. HO(G; A) HO(G; Al) since the action of G is ergodic. 1.0. THEOREM. The long exact sequence (cf. [3, page 116]) 0 -HO (G; A1/A) . -> Hn(G; A) ->Hn(G; A1) -+Hn(G; A1/A) -*Hn+ (G; A) *. . is isomorphic to the long exact sequence for h: 5-*? as given in [4, 4.1 1], with A as the coefficient group. In particular, Hn(5Y; A) _LHn(G; A,) and Hn+'(h; A)-Hn(G, A /A), for n>0. PROOF. We will write elements of af as (x, y) instead of (x, 7, 'y-1 x), Received by the editors August 7, 1969. AMS 1969 subject classifications. Primary 2870, 2050, 2095.

It is shown in this paper that if G is a finite group which is the product of two normal supersolvable subgroups of relatively prime index, then G is supersolvable. In 1953, Huppert [1] gave an example showing that, in general, the product of two normal supersolvable subgroups of a finite group need not be supersolvable. The purpose of this note is to show that if the subgroups have relatively prime index in their product, then the product is again supersolvable.2 THEOREM. Let A, B be normal supersolvable subgroups of the finite group G with G = AB. If the indices [G: A ] and [G: B ] are relatively prime, then G is supersolvable. PROOF. Assume that the theorem is false, and that G is a counterexample of minimal order. Thus we may assume that the indices of A and B in G are distinct primes p and q respectively with A/A CB cyclic of order q and B/AnB cyclic of order p. Let a(AnB) and i3(A CnB) generate these quotients and assume that a has order q and j has order pt for some s, t > 1. Let E1 be a minimal normal subgroup of A contained in ACOB. If AOB is trivial, the result follows immediately so E1 is nontrivial and has order a prime r. If E1 is normal in G, then by the minimality of G, we must have G/E1 supersolvable; and since E1 is cyclic, G itself must be supersolvable. Thus E1 has p distinct conjugates E1, E2, * * *, E.. Then each Ei< A and hence E = (E1, E2, * * *, Ep), the group generated by the Ei is an elementary abelian r-group of order

In this paper all finite groups having exactly one irreducible K-representation of degree greater than one are determined, where K is an algebraically closed field of characteristic zero. The quaternion group of order eight and the dihedral group of order eight are nilpotent groups with this property, while the symmetric group on three letters and the alternating group on four letters are solvable although not nilpotent examples. The theorem below will show how typical the above examples really are. In the following all groups are finite. If G is a group, let GI denote the derived group of G and Z(G) the center of G. Let K be an algebraically closed field of characteristic zero.

and posed the question of determining whether or not A(m) contains a principal congruence subgroup for larger m. This question was settled in the negative for m not a prime power by I. Reiner in his paper [4], and in [3] M. I. Knopp proved that the groups A(m) are of infinite index in r for m ? 6 and so cannot possibly contain a principal congruence subgroup, these being of finite index always. Knopp also points out that the question had been considered and settled by F. Klein and R. Fricke in their treatise on the modular functions. In this note we wish to point out that A (6) is an interesting group. In fact we prove

The purpose of this paper is to rephrase a conjecture about simple groups into the language of linear algebra. Let G be a group of finite order o(G). Then by rF we shall mean the group ring of G over a field of characteristic p (for instance the integers modulo p). We shall denote the radical of rF by N,. If p = 0 or p o(G), then it is known that Np=(O); and if p|o(G), Np (O). We now consider the following two assertions: (A) If G is a simple group of odd order, o(G) is a prime. (B) If G is a group of odd order o(G), then for some prime p, p[ o(G), we can find a gCG, g1, such that g-1CNp. The theorem which we propose to prove is

THEOREM. Let G be any locally compact nondiscrete group (or any infinite discrete amenable group). Then the radical of the (complex, noncommutative) Banach algebra LOO(G)* is not norm separable. Introduction. Let G be a locally, compact group and denote by 9 the radical of the complex Banach algebra Lm(G)* with Arens multiplication induced from L'(G) (see next section). It has been proved by Civin and Yood [2] that if G is nondiscrete locally compact abelian (or G is the group of additive integers) then R is infinite dimensional. The conjecture of [2] that this holds true for all discrete abelian G has been proved by this author in [5, pp. 48, 58]; and, in fact, for any discrete infinite amenable group G, M. is infinite dimensional [5]. It has been later proved by S. Gulick in [8, p. 136], among other results, that if G is any locally compact abelian group then 9 is not even norm separable. The following is posed as an unsolved question in [8, p. 136]: "If G is an infinite nonabelian locally compact group is the radical of L>(G)* nonzero?" The theorem quoted in the abstract gives an incomplete answer to this question. The case it leaves out is for G an infinite discrete nonamenable group, for which the methods used in this paper fail. The method of proof resembles the one used by Gulick. We keep, however, a much closer account of the items involved. Definitions and notations. Unless otherwise stated we adhere to the definitions of Hewitt-Ross [9]. C(G) [UCBr(G)] will denote the space of bounded [right uniformly] continuous, complex, bounded functions with usual sup norm (f e UCBr(G) iff f e C(G) and the map x-df, 1fJ(y)=f (xy), from G to C(G) is continuous). Thus UCBr(G)C C(G)c L (G). The Arens multiplication is defined by: If u, v c LX (G), f e LI(G), g EL1(G)thenf o g EL'(G),v o fE LO(G), 0o vE L"(G)* aredefined by: (f og) ()=(f,g* ), (vof) ()=v(f 0) for all eL1(G) and Received by the editors October 6, 1972. AMS (MOS) subject class.fications (1970). Primary 22D 15, 22D05, 43A 10; Secondary 46H05, 43A15, 43A07. I Research done while the author held a Canada Council Award. The support of the Canada Council is gratefully acknowledged. ? American Mathematical Society 1973

This paper studies the derivations and automnorphisms of the incidence algebra of a locally finite partially ordered set. Two subspaces are shown to span the space of derivations: the space of inner derivations and the space of derivations associated with the additive functions. An analogous result is shown for the group of automorphisms. A number of dimension calculations are also made. 1. Preliminaries. In the study of additive and multiplicative functions on the segments of a (partially) ordered set, it is natural to consider derivations and automorphisms, respectively, of the incidence algebra. In ?2, we study the derivations. In ?3, we give the analogous results for automorphisms. For a more complete description of the notation used see [1]. Let P be a locally finite (partially) ordered set, i.e. for which every segment [x,y]={z|x

The number of isotype subgroups of an abelian p-group G is determined. This solves a recent problem of Fuchs. Actually, we accomplish slightly more. Define a subgroup H of an abelianp-group G to be an i-pure subgroup of G if, for some ordinal A, H is pA-pure in G and pAH is divisible. We compute the number of i-pure subgroups of G and show that the number of i-pure subgroups and the number of isotype subgroups of G coincide. Our final result deals with the number of nonisomorphic isotype subgroups of G when G is a direct sum of countable groups. In this paper all groups are abelian, and "abelian group" is shortened to "group". For the most part, our notation and terminology agree with [2]. However, for the convenience of the reader we list the meaning of the following frequently used symbols and terms.

In this paper, we shall be concerned with the question of what conditions on minimal transformation groups will guarantee that they are disjoint. Generalizing a result of I. Bronstein about lifting of minimality through group extensions to associated bitransformation groups, we prove that in a large class of transformation groups, disjointness is equivalent to disjointness of their maximal equicontinuous factors. In the abelian case, this means that disjointness is equivalent to no common factor in the class of flows discussed.