In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). If X is such a graph with vertex set V(X) and automorphism group A(X), we say that X is a graphical regular representation (GRR) of a given abstract group G if

(I) G ≅ A(X) , and

(II) A(X) acts on V(X) as a regular permutation group; that is, given u, v ∈ V(X), there exists a unique φ ∈ A(X) for which φ(u) = v.

That for any abstract group G there exists a graph X satisfying (I) is well-known (cf. [3]).

The present paper is a sequel to the previous paper bearing the same title by the same authors [3] and which will be hereafter designated by the bold-face Roman numeral I. Further results are obtained in determining whether a given finite non-abelian group G has a graphical regular representation. In particular, an affirmative answer will be given if (|G|, 6) = 1.

Inasmuch as much of the machinery of I will be required in the proofs to be presented and a perusal of I is strongly recommended to set the stage and provide motivation for this paper, an independent and redundant introduction will be omitted in the interest of economy.

This paper develops some techniques for the study of derivation algebras and cohomology groups of Lie algebras. We are especially concerned with solvable algebras over arbitrary fields with structural properties like those of the Borel subalgebras of complex semi-simple Lie algebras. In particular, these algebras are semi-direct sums of nilpotent ideals and abelian subalgebras which act on the ideals in a semi-simple fashion. We make strong use, in our discussion, of a cohomology theorem of Hochschild-Serre. This result is stated herein (§ 2) in a modified form which allows us to omit the original hypothesis that the base field have characteristic 0.

A Stone lattice is a distributive, pseudo-complemented lattice L such that a* V a** = 1, for all a in L; or equivalently, a bounded distributive lattice L in which, for all a in L, the annihilator a⊥ = {b ∊ L|a ∧ b = 0} is a principal ideal generated by an element of the centre of L, namely a*.

Thus it is natural to define an 𝔪-Stone lattice to be a bounded distributive lattice L in which, for each subset A of cardinality less than or equal to m, the annihilator A⊥ = {b ∊ L|a ∧ b = 0, for all a ∊ A} is a principal ideal generated by an element of the centre of L.

In this paper we characterize 𝔪-Stone lattices, and show, by considering the lattice of global sections of an appropriate sheaf, that any bounded distributive lattice can be embedded in an 𝔪-Stone lattice, the embedding being a left adjoint to the forgetful functor.

In [1] Arhangel'skiĭ announced that any σ-paracompact p-space could be mapped onto a Moore space by a perfect map. However Burke [3] recently showed that this is not true in general and he gave an example of a T2, locally compact, σ-paracompact space which cannot be mapped onto a Moore space by a perfect map.

In [4], Hofmann defines a locally compact group with zero as a Hausdorff locally compact topological semigroup, S, with a non-isolated point, 0, such that G = S — {0} is a group. He shows there that 0 is indeed a zero for 5, G is a locally compact topological group, and the identity of G is the identity of S. The author has investigated actions of such semigroups on locally compact spaces in [1; 2]. In this paper, we are investigating direct products of semigroups of the above type and actions of these products; for a special case of this, the reader is referred to [3].

Copeland [1] proved that if X is a CW complex then the set of homotopy classes of multiplications is in 1-1 correspondence with the loop [X ∧ X, X], But in general [X ∧ X, X] is very hard to compute.

Here we study the problem of finding how many different (up to homotopy) multiplications can be put on a space with finitely many non-vanishing homotopy groups. We reduce this problem to computing quotients of certain cohomology groups and to determining the primitive elements (Theorem 3.6). Our approach to this problem uses Postnikov decompositions and multiplier arguments.

Using ultraproducts, N. R. Reilly proved that if G is a representable lattice-ordered group and J is an independent subset totally ordered by ≺, then the order on G can be extended to a total order which induces ≺ on J (see [5]). In [4], H. A. Hollister proved that a group G admits a total order if and only if it admits a representable order and, moreover, every latticeorder on a group is the intersection of right total orders. The purpose of this paper is to provide a partial converse, viz: if G is a lattice-ordered group and J is an independent subset totally ordered by ≺, then the order on G can be extended to a right total order which induces ≺ on J.

By “algebra” we shall mean a finitary universal algebra, that is, a pair 〈A; F〉 where A and F are nonvoid sets and every element of F is a function, defined on A, of some finite number of variables. Armbrust and Schmidt showed in [1] that for any finite nonvoid set A, every group G of permutations of A is the automorphism group of an algebra defined on A and having only one operation, whose rank is the cardinality of A. In [6], Jónsson gave a necessary and sufficient condition for a given permutation group to be the automorphism group of an algebra, whereupon Plonka [8] modified Jonsson's condition to characterize the automorphism groups of algebras whose operations have ranks not exceeding a prescribed bound.

In the well-known solution to Post's problem, Friedberg [1] and Muchnik [13] each constructed a pair of incomparable recursively enumerable (r.e.) degrees a and b. Subsequently, Sacks [15, p. 81] constructed r.e. degrees c and d such that c ∪ d = 0’ and C’ = d’ = 0'. Lachlan showed [7, p. 69] that such degrees c, d could have no greatest lower bound in the upper semilattice of r.e. degrees. We show that the original Friedberg-Muchnik degrees a, b automatically satisfy Sack's conditions and hence witness that the upper semi-lattice of r.e. degrees is not a lattice.

Given a class of topological spaces and a class of maps of topological spaces, our interest is in characterization of the class () of topological spaces whose every -image lies in . The class () is referred to as the -resolvent of and is the largest class of spaces smaller than closed under -images (provided is closed under composition and includes identity maps).

In the present paper, will be either the class of paracompact spaces or the class of regular spaces, and the conditions determining will always include separation axioms on the range, some of which can be dispensed with now by agreeing that all spaces are Hausdorff.

Let S(≠1) be a subgroup of a group G. We consider the question: when are the conjugates of S “as independent as possible“? Specifically, suppose SG (the normal subgroup generated by S in G) is the free product II*S0α where and gα ranges over a subset J of G. Then J must be part of a (left) coset representative system for G mod SG. N where N is the normalizer of S in G. (For, g ∊ SGgαN implies Sg is conjugate to Sgα in SG; however, distinct non-trivial free factors of a free product are never conjugate.)

We say that SG is the free product of maximally many conjugates of S in G if SG = II*Sgα where gα ranges over a (complete) left coset representative system for G mod SGN (or equivalently, gα ranges over a double coset representative system for G mod (SG, N)); in this case we say briefly that S has the fpmmc property in G.

One of the most fundamental and characteristic features of recursion theory is the fact that the recursive sets are not uniformly recursive. In this paper we consider the degrees a such that the recursive sets are uniformly of degree ≦a and characterize them by the condition a’ ≦ 0". A number of related results will be proved, and one of these will be combined with a theorem of Yates to show that there is no r.e. degree a < 0’ such that the r.e. sets of degree ≦a are uniformly of degree ≦a. This result and a generalization will be used to study the relationship between Turing and many-one reducibility on the r.e. sets.

An h by h matrix with entries ±1 is called a modular Hadamard matrix if the inner product of any two distinct row vectors is a multiple of a fixed (positive) integer n; such a matrix is also referred to as an “H(n, h) matrix” with parameters n and h. Modular Hadamard matrices and the related combinatorial designs were introduced in [2]; that paper was concerned mainly with two of the related designs, the “pseudo (ν, k, λ)- designs” and the “ (m, v, k1, λ1, k2, λ2, f, λ3)-designs” (the reader is referred to [2] for the definition of these designs).

Throughout this paper V will denote an Archimedean Riesz space with a weak unit e and a zero element θ. A sequence f1,f2,f3, … of points of V is said to converge relatively uniformly to a point f (with regulator the point g of V) if, for each ∈ > 0, there is a number N such that, if n is a positive integer and n > N, then |f — fn| < ∈g. In an Archimedean Riesz space a relatively uniformly convergent sequence has a unique limit. The sequence f1, f2, f3, … is called a relatively uniform Cauchy sequence (with regulator g) if, for each ∈ > 0, there is a number N such that if n and m are positive integers and n, m > N, then |fn — fm| < eg. A subset M of V is said to be sequentially relatively uniformly complete, written s.r.u.-complete, whenever every relatively uniform Cauchy sequence of points of M (with regulator in V) converges to a point of M. This property was defined by Luxemburg and Moore in [4] and some related conditions were derived.

Anosov flows are a generalization of geodesic flows on the unit tangent bundles of compact manifolds of negative sectional curvature. They were introduced and are dealt with at length by Anosov in [2]. Moreover they form an important class of examples of flows satisfying Smale's axioms A and B (see [15]). In that paper Smale poses the problem of determining which manifolds admit Anosov flows. In this paper we obtain information about the fundamental groups of such manifolds. These generalize results which have been obtained for the fundamental groups of manifolds of negative curvature (see Preissmann [13], Byers [6]).

In [1; 2 ; 7] Gabriel, Goldman, and Silver have introduced the notion of a localization of a ring which generalizes the usual notion of a localization of a commutative ring at a prime. These rings may not be local in the sense of having a unique maximal ideal. If we are to obtain information about a ring R from one of its localizations, Qτ (R) say, it seems reasonable that Qτ(R) be a tractable ring. This, of course, is what Goldie, Jans, and Vinsonhaler [4; 3; 8] did in the special case for Q(R) the classical ring of quotients.

In this paper, we refine the presentations of Behr and Mennicke [1] for SL(2, m) and PSL(2, m) where m is odd. The group SL(2, m) is first shown to be presented by the following system of generators and relations:

1.1

The group PSL(2, m) appears as the factor group

1.2

This simplification then permits us to use the results of Schur [3] to establish three-relation presentations for these groups.

One of the most useful properties of a compact Hausdorff space is that such a space is closed whenever embedded into a Hausdorff space. This property does not extend to compact spaces with respect to embeddings into arbitrary spaces. Thus, an interesting topological problem is to characterize the types of absolute “closure” properties that are possessed by compact spaces. This is the problem that is solved in the present paper.

The following notation and terminology will be used below. We shall consider a fixed space X and subspace A, representing arbitrary nonempty open subsets of X (respectively A ) by W (respectively V).

We denote by ZG the integral group ring of the finite group G. S.D. Berman [1] showed that every unit of finite order μ in G is trivial (i.e., μ = ±g for some g in G) if and only if either G is abelian or G is a Hamiltonian 2-group. In this note, we give a new and shorter proof for the “only if” part.