The object of this note is to draw attention to a family of eight uniform compounds of regular polyhedra which arise out of an action of the group on the regular icosahedron.

Let R be a commutative ring. We say that R satisfies the ascending chain condition for principal ideals, or that R has property (M), if each ascending sequence (a1) ⊆ (a2) ⊆ … of principal ideals of R terminates. Property (M) is equivalent to the maximum condition on principal ideals; that is, under the partial order of set containment, each collection of principal ideals of R has a maximum element. Noetherian rings, of course, have property (M), but the converse is not true; for if R has property (M) and if {Xλ} is a set of indeterminates over R, then the polynomial ring R[{Xλ}] has property (M). Krull domains, and hence unique factorization domains, have property (M).

1.1. Introduction. In this paper, we continue with the theme of (1): the relationships holding between the Sπ (i.e. maximal π) subgroups of a locally finite group and the various local systems of that group. In (1), we were mainly concerned with ‘good’ Sπ subgroups – those which reduce into some local system (and are said to be good with respect to that system). Here, on the other hand, we are concerned with a very much more special sort of Sπ subgroup.

We recall from (3) that a group G is (centrally) eremitic if there exists a positive integer e such that, whenever an element of G has some power in a centralizer, it has its eth power. The eccentricity of an eremitic group G is the least such positive integer e.

In ((4), Theorem A) we proved that if A is a torsion free Abelian normal subgroup of a finitely generated group G with G/A nilpotent, then G has a subgroup of finite index with eccentricity 1. In this note we use a much simpler method to prove a stronger result.

Let A1 denote the algebra generated by one pair of canonical operators. It is shown that the realizations of sl(2) in A1 divide into two non-empty classes determined by the spectrum of the semisimple element of sl(2) in A1.

1. When, as will be presumed henceforward, no two of a0, a1, …, an are equal the n + 1 equations

are linearly independent; x0, x1, …, xn are homogeneous coordinates in [n] projective space of n dimensions—and the simplex of reference S is self-polar for all the quadrics.

In this article we study certain subclasses of the class ℒ of Lagrangian groups; that is, finite groups G having, for every divisor d of |G|, a subgroup of index d. Two such subclasses, mentioned by McLain in (6), are the class ℒ1 of groups G such that every factor group of G is in ℒ, and the class ℒ2 of groups G such that each subnormal subgroup of G is in ℒ. In section 1 we prove that a group of odd order in ℒ1 is supersoluble, and give some examples of non-supersoluble groups in ℒ1. Section 2 contains several results on the class ℒ2. In particular, it is shown that a group in ℒ2 has an ordered Sylow tower and, after constructing some examples of groups in ℒ2, a result on the rank of a group in ℒ2 is proved (Theorem 4).

The following extension problem is of considerable geometrical interest. Given a compact manifold X = ∂Z the boundary of a compact manifold Z, does the normal bundle of X in Z extend to a line bundle on Z? A related question involving foliations is if X is a compact boundary, is X a leaf of a codimension one foliation on any compact manifold bounded by X? Obviously if it were such a leaf, its normal bundle in the bounded manifold would extend. Reinhart posed both these problems in the language of cobordism theories(1), (2) defining a vector cobordism theory to a deal with questions of the first genre and a foliated-cobordism theory to deal with the second. He was able to obtain complete results in his vector-cobordism theory, whilst the foliated-cobordism theory remains an unsolved problem. Our aim in this article is to extend Reinhart's results on vector-cobordism theory to unoriented line bundles. Whereas in vector-cobordism the Euler number is a cobordism invariant, in the cobordism theory that we shall describe, the modulus of the Euler number becomes a cobordism invariant.

Let S be a finite p-group and let σ be a p'-automorphism of S. A well-known result of Huppert ((2), IV, (5·12), Satz) states that if σ acts trivially onΩ1(S) (ω2(S) if p = 2) then σ is the identity. In this paper, we give a short proof of this result and also give some applications of it. Our proof depends on a somewhat surprising Lemma (Lemma 1) which is of some interest in its own right.

The modular group, Γ, is the group of linear fractional transformations with integral coefficients and determinant 1. The group is generated by the transformations

which have orders 2 and 3 respectively. The transformation

is of infinite order. Abstractly, the group can be viewed as the free product of cyclic groups of order 2 and 3.

Introduction. The space Cp is the class of operators on a Hilbert space for which the norm ∥K∥p = [trace (KK*)P/2]/p is finite. Equivalently, a compact operator is in Cp if

where the μn are the so-called ‘singular values’ of K (characteristic values of the non-negative compact operator [K] ≡ (KK*)½). The case p = 2 gives the familiar class of Hilbert–Schmidt operators, while C1 is the collection of trace-class or nuclear operators considered by Schatten(12), Lidskii(11), and Gohberg and Krein(7), among others.

0. Introduction. A group π has weak dimension (wd) ≤ n (see Cartan and Ellen-berg (2)) if Hk(π, A) = 0 for all right Z(π)-modules A and all k > n. We say that the weak dimension of a manifold M is ≤ n if wd (πl(M))≤ n. In section 1 we show that open, orientable, irreducible 3-manifolds have wd ≤ 1 if and only if they are the monotone on of 1-handle bodies. In his celebrated theorem (10), Stallings proves that finitely presented groups of cohomological dimensions ≤ 1 are free. In section 2 we prove that if π is a finitely presented group which is the fundamental group of any orientable 3-manifold with wd ≤ 1 then π is free. We also give an example to show that the finite generation of π is necessary. (Swan (11) removes the finitely presented hypothesis from Stalling's theorem.) Finally, in section 3 we generalize a theorem of McMillan (5) and prove that if M is an open, orientable, irreducible 3-manifold with finitely generated fundamental group, then M is stably (taking the product with n ≥ 1 copies of ℝ) a connected sum along the boundary of trivial (n+2)-disc Sl bundles.

In (1) (where references and terminology may be found), C. A. Rogers has proved the following surprising theorem.

Letbe a class of sets, all Souslin-in a topological space X. If A is a Souslin-set and B is an analytic set that does not meet A, then there is a Borelian-set C with A ⊂ C and C ∩ B = Ø

A Petrie matrix (Kendall (1), (2)) is a matrix of O's and l's such that the l's in each column occur consecutively. An example follows.

Introduction. All the sets X considered in this paper will be compact convex subsets of the Euclidean plane R. We shall require the following definitions. For x, y, z ∈ R let L(x, y, x) denote the minimum of all the lengths of all connected sets containing x, y and z. Define the stretch of x with respect to X to be

and the stretch of X to be

The following Conjecture was made in (6).

CONJECTURE A. Let Xn be a closed manifold of type K(π, 1). Then X˜n is homeornorphic to.

In section 4, we prove the conjecture when n ≥ 5 and π is a non-trivial direct product. More general than Conjecture A would be

CONJECTURE B. Let Xn be a manifold of type K(π, 1) with ∂X = Ø. Then X˜n is homeomorphic to.

In section 4 we give an example to show that Conjecture B is false in each dimension n ≥ 4.

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.

Let X1, X2, … be a sequence of independent random variables such that, for each n ≥ 1, EXn = 0 and and assume that then converges almost surely as N → ∞. Let and let Fn(x) denote the distribution function of Xn. Loynes (2) observed that the sequence {Sn} is a reversed martingale, and applied his central limit theorem to it: however, stronger results are obtainable, in precise duality with the classical theory of partial sums of independent random variables. These results describe the fluctuations of the sequence {Sn}, and hence the way in which converges to its limit.

Kirby and Siebenmann (in (7)) proved the annulus conjecture in dimensions five and above, by proving the stable homeomorphism conjecture in those dimensions, that is, by proving that every orientation-preserving homeomorphism of Rn is stable, if n ≥ 5. Although this result apparently lessens the importance of stable homeomorphisms, the concept of stability should still be of use in settling the fate of the annulus conjecture in dimension 4, or for uncovering a simple (i.e. non-surgical) proof in other dimensions.

1. Introduction. We shall employ the customary abbreviations: p.d.f. for probability density function; d.f. for distribution function; c.f. for characteristic function.