bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the Decomposition of Groups
Authors: Hill Paul 
Published: December 2010

The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property.Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G.Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.
The Distribution of Primitive Roots
Authors: Elliott P. D. T. A. 
Published: December 2010

Notation,p and q are generic symbols for prime numbers. N(H, p) denotes the number of primes q, not exceeding Hy which are primitive roots (mod p).g(p) denotes the least positive primitive root (mod p).g*(p) is the least prime primitive root (mod p).v(m) denotes the number of distinct prime divisors of the integer m.τk(m) is the number of ways of representing the integer m as the product of k integers, order being important.π(x, k, r) is the number of primes p, not exceeding x, which satisfy p ≡ r(mod k); while π(x) denotes the total number of p ≦ x.logm x denotes the mth iterated logarithmic function which is defined, when possible, by
Federer-Čech Couples
Authors: Dyer Micheal 
Published: December 2010

In (5),I considered two-term conditions in π-exact couples, of which the exact couple of Federer (7) is an example. Let M(X, Y)be the space of all maps from X to Y with the compact-open topology. Our aim in this paper is to construct a π-exact couple , where Xis a finite-dimensional (in the sense of Lebesgue) metric space and , a certain (rather large) class of spaces. Specifically, is the class of all topological spaces Xwhich possess the following property (P).(P) Let Y be a (possibly infinite) simplicial complex. There exists x0 ∈ X and y0 ∊ Y such that [X, x0]≃ [Y, y0].In § 5 it will be seen that contains all CW complexes and all metric absolute neighbourhood retracts (ANR)s.
Axioms for Absolute Geometry. II
Authors: Rigby J. F. 
Published: December 2010

In this paper I continue the process, begun in (2), of reducing and weakening the axioms of congruence needed for absolute geometry. The congruence axioms Cl*–C4*, C4**, and C5a–C5c (frequently referred to below) can all be found in (2) and will not be quoted again here. (This paper should be read in conjunction with (2); any attempt to make it self-contained would result in the repetition of large parts of (2).) The notation of (2) will be used throughout the paper.The main result here is that axiom C5c is unnecessary. This is shown in § 1. In § 2 I discuss three other points arising from (2).Note added in proof.Since writing this paper, I have constructed examples of (a) Archimedean planes satisfying Cl*-C4* in which not all points are isometric, (b) non-Archimedean planes satisfying Cl*-C4*but not C4**,and (c) one-dimensional geometries in which 2.1 (with “plane” replaced by “line“) is false.
Nil Subrings of Goldie Rings are Nilpotent
Authors: Lanski Charles 
Published: December 2010

Herstein and Small have shown (1) that nil rings which satisfy certain chain conditions are nilpotent. In particular, this is true for nil (left) Goldie rings. The result obtained here is a generalization of their result to the case of any nil subring of a Goldie ring.Definition. Lis a left annihilator in the ring R if there exists a subset S ⊂ R with L = {x∈ R|xS= 0}. In this case we write L= l(S). A right annihilator K = r(S) is defined similarly.Definition. A ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilators terminates at some point. We recall the well-known fact that this condition is inherited by subrings.Definition. R is a Goldie ring if R has no infinite direct sum of left ideals and has the ascending chain condition on left annihilators.
On the Automorphisms of Infinite Chevalley Groups
Authors: Humphreys J. E. 
Published: December 2010

In (8, § 3.2) Steinberg proved the following result.THEOREM. Let K be a finite field, G′ a simple Chevalley group (“normal type1”) over K. Then every automorphism of G’ is the composite of inner, graph, field, and diagonal automorphisms.For the meaning of these notions, see (8). Our aim in this note is to indicate how the Theorem may be extended to arbitrary infinite fields K, provided we replace G′ by the group denoted G in (5) and Ĝ in (8). This amounts to proving the Theorem for automorphisms of G′ which are induced by automorphisms of G; when K is finite, Steinberg's results show that all automorphisms of G′ arise in this way. As Steinberg points out, the sole use made of the finiteness of K in his argument is in the proof of the following statement: Let U be the subgroup of G′ corresponding to the set of positive roots, and let σ be any automorphism of G′; then Uσ is conjugate to U in G′.
Graphs with Maximal Even Girth
Authors: Gewirtz A. 
Published: December 2010

In this paper we examine the classGof simple undirected, connected graphs of diameterd> 1, girth2d,and for anygɛG, if a pair of nodes are at distancedfrom each other, then that pair of nodes is connected bytdistinct paths of lengthd, t> 1. (The girth ofgis the length of the smallest circuit ing.)We establish, in § 2, that for allgɛG, gis regular.We establish necessary conditions for the existence of elements ofG.IfgɛG,we adopt the notationg=g(d, t, v, n),wherevis the valence ofgandnis the number of nodes. It is of course possible forg, hɛG, g ≠ h,and for givend, t, v, nto have bothg(d, t, v, n)andh(d, t, v, n).In particular, we show that ifd= 2,t≠ 2, 4 or 6, then there is at most a finite number of graphs with a particular giventvalue.
Finite Linear Groups of Prime Degree
Authors: Wales David B. 
Published: December 2010

If G is a finite group which has a faithful complex representation of degree nit is said to be a linear group of degree n. It is convenient to consider only unimodular irreducible representations. For n ≦ 4 these groups have been known for a long time. An account may be found in Blichfeldt's book (1). For n= 5 they were determined by Brauer in (4). In (4), many properties of linear groups of prime degree pwere determined for pa prime greater than or equal to 5.In a forthcoming series of papers these results will be extended and the linear groups of degree 7 determined. In the first paper, some general results on linear groups of degree p, p≧ 7, will be given. These results will later be
applied to the prime p = 7.
A Note on the Construction of Projective Planes from Groups
Authors: Kirkpatrick P. B. 
Published: December 2010

André (1) gave a construction for translation planes from abelian groups possessing “congruences” of subgroups. Schwerdtfeger (3) constructed the plane over a field F from the group of substitutions x → ax + b (a, b ∊ F; a ≠ 0). In this note we describe a construction (inspired by Schwerdtfeger's work), from groups, of planes which are duals of near-field planes.If a plane is (l, m)-transitive (cf. 2, p. 67) for some pair of distinct lines l, m, then the central collineations ϕ with axis m and centre on I may be identified with the “proper” points (that is, points not on I or m) of the plane once an origin O is chosen (not on l or m):
On the Semisimplicity of Modular Group Algebras. II
Authors: Passman D. S. 
Published: December 2010

Let G be a discrete group, let Kbe an algebraically closed field of characteristic p > 0 and let KGdenote the group algebra of Gover K.In a previous paper (2) I studied the Jacobson radical JKGof KGfor groups Gwith big abelian subgroups or quotient groups. It is therefore natural to next consider metabelian groups, and I do this here. The main result is as follows.THEOREM 1. Let K be an algebraically closed field of characteristic p and let a group G have a normal abelian subgroup A with G/A abelian. Then JKG ≠ {0} if and only if G has an element g of order p such that the A-conjugacy class gA is finite and such that the group is periodic.Note that since and G/Ais abelian, we do in fact have .
On the Isomorphism of Integral Group Rings. II
Authors: Sehgal Sudarshan K. 
Published: December 2010

Let Z(G) denote the integral group ring of a group G. Let be the class of groups G with the property that for any isomorphism θ: Z(G) → Z(H), we have θ(g) = ±h, h ∈ H, for all g ∈ G. We study this class in § 2 and establish that it contains classes of torsion-free abelian groups, torsion abelian groups, and ordered groups.In § 4, we prove the following result.THEOREM. Let G be a group which contains a normal abelian subgroup A such that. Suppose that θ: Z(G) → Z(H) is an isomorphism such that θ(Δ(G, A)) = Δ(H, B) for a suitable normal subgroup B of H. Then G ≃ H. (Here Δ(G, A) is the kernel of the natural map Z(G) → Z(G/A).)Jackson (3) and Whitcomb (6) proved the special case of this theorem when G is supposed to be finite metabelian. The lemmas needed are given in §3.
On the Cones Associated with Biorthogonal Systems and Bases in Banach Spaces
Authors: Mcarthur C. W. , Singer Ivan , Levin Mark 
Published: December 2010

1. Let E be a Banach space (by this we shall mean, for simplicity, a real Banach space) and (xn,fn) ({xn} ⊂ E, {fn} ⊂ E*) a biorthogonal system, such that {fn} is total on E (i.e. the relations x ∈ E,fn(x) = 0, n = 1, 2, …, imply x = 0). Then it is natural to consider the cone1which we shall call “the cone associated with the biorthogonal system (xn,fn)”. In particular, if {xn} is a basis of E and {fn} the sequence of coefficient functional associated with the basis {xn}, this cone is nothing else but2and we shall call it “the cone associated with the basis {xn}”.
Reconstruction of Cacti
Authors: Geller Dennis , Manvel Bennet 
Published: December 2010

Following the work of Kelly (8), Harary and Palmer (5), and Bondy (1) on the reconstruction of trees, and of Manvel (10) on the reconstruction of connected graphs with a single cycle, it was a natural step to attempt to solve the reconstruction problem for cacti. The solution of this problem, presented here, assumes both Kelly's Theorem and the result of Manvel in (10). Any definitions not given here can be found in (2).Let graph G have point set V = {v1 v2, …, vp} and line set X = {x1, x2, …, xq}. For each vi ∈ V, Gi = G – vi is the maximal subgraph of G which does not contain vi and is formed by deleting vi and all lines incident with it from G.
Generalized Discrete Valuation Rings
Authors: Brungs H.-H. 
Published: December 2010

Jategaonkar (5) has constructed a class of rings which can be used to provide counterexamples to problems concerning unique factorization in non-commutative domains, the left-right symmetry of the global dimension for a right- Noetherian ring and the transhnite powers of the Jacobson radical of a right- Noetherian ring. These rings have the following property:(W) Every non-empty family of right ideals of the ring R contains exactly one maximal element.In the present paper we wish to consider rings, with unit element, which satisfy property (W). This property means that the right ideals are inverse well-ordered by inclusion, and it is our aim to describe these rings by their order type. Rings of this kind appear as a generalization of discrete valuation rings in R; see (1; 2).In the following, R will always denote a ring with unit element satisfying (W).
Euclidean Lie Algebras
Authors: Moody Robert V. 
Published: December 2010

Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E((A ij)) is simple providing that (A ij) does not fall in the list given in (4, Table). We did not prove the converse, however.The diagrams of the table of (4) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.
Distinct Values of a Polynomial in Subsets of a Finite Field
Authors: Williams Kenneth S. 
Published: December 2010

If A is a set with only a finite number of elements, we write |A| for the number of elements in A. Let p be a large prime and let m be a positive integer fixed independently of p. We write [pm] for the finite field with pm elements and [pm]′ for [pm] – {0}. We consider in this paper only subsets H of [pm] for which |H| = h satisfies1.1If f(x) ∈ [pm, x] we let N(f; H) denote the number of distinct values of y in H for which at least one of the roots of f(x) = y is in [pm]. We write d(d ≥ 1) for the degree of f and suppose throughout that d is fixed and that p ≧ p0(d), for some prime p0, depending only on d, which is greater than d.
Differentiable Montgomery-Samelson Fiberings with Finite Singular Sets
Authors: Antonelli Peter L. 
Published: December 2010

In 1946 Montgomery and Samelson (11) introduced a generalization of the notion of a differentiable group action with one type of orbit besides fixed points. Such an object is essentially a locally trivial fibering except on a certain singular set over which fibres are pinched to points. In recent years there has been a fair amount of research on these MS-fiberings and similar singular fiberings. This paper is another effort in this direction. For a fairly complete bibliography of the literature, the reader should consult the references, and in particular, (5).Let f: Mn → Sp, with Mn a closed connected n-manifold and Sp the unit p-sphere with standard differentiable structure, be the projection map of a smooth MS-fibering with finite non-empty singular set.
Notes on Locally Compact Connected Topological Lattices
Authors: Choe Tae Ho 
Published: December 2010

I t was shown in (2) that if(1) L is a locally compact connected topological lattice and if(2) L is topologically contained in R2, the Euclidean plane, then each compact subset of L has an upper bound and a lower bound in L. I t was also asked whether this result could be proved without assuming condition (2). In this note, we show that this result continues to hold if condition (2) is weakened to: L is finite-dimensional.In (11), it was shown that the centre of a compact topological lattice is totally disconnected. We shall prove t h a t this result is also true even in a locally compact, locally convex topological lattice with 0 and 1. This yields that any locally compact topological Boolean algebra is totally disconnected.
Residually Finite Rings
Authors: Chew Kim Lin , Lawn Sherry 
Published: December 2010

Throughout this paper a ring will always be an associative, not necessarily commutative ring with an identity. It is tacitly assumed that the identity of a subring coincides with that of the whole ring. A ring R is said to be residually finite if it satisfies one of the following equivalent conditions:(1) Every non-zero ideal of R is of finite index in R;(2) For each non-zero ideal A of R, the residue class ring R/A is finite;(3) Every proper homomorphic image of R is finite.The class of residually finite rings is large enough to merit our investigation. All finite rings and all simple rings are trivially residually finite. Other residually finite rings are said to be proper.
On Absolute Summability by Riesz and Generalized Cesàro Means. II
Authors: Körle H.-H. 
Published: December 2010

1. We will use the terminology of part I [9], including the general assumptions of [9, § 1]. In that paper we had proved that |R, λ, κ| = |C, λ, κ| in case that κ is an integer. Now, we turn to non-integral orders κ.As to ordinary summation, the following inclusion relations (in the customary sense; see [9, end of § 1]) for non-integral κ have been established so far. (Since we are comparing Riesz methods of the same type λ and order κ only, (R, λ, κ) is written (R), etc., for the moment.) (R) ⊆ (C) is a result by Borwein and Russell [2]. (C) ⊆ (R) was proved by Jurkat [3] in the case 0 < κ < 1, and, after Borwein [1], it holds in the case 1 < κ < 2 if(1)(2)(i.e. decreases in the wide sense).
Extensions of Certain Maps to Automorphisms of m
Authors: Berg I. D. 
Published: December 2010

In this paper we consider Banach space automorphisms of m, the space of bounded sequences, which map c, the space of convergent sequences, into itself. In particular, we consider the problem of determining which maps from C0, the space of sequences converging to 0, to c can be extended to such automorphisms.The origin of this note lies in an incorrect conjecture of mine. If the automorphism T: m → m is given by a matrix, that is, a sequence of elements of ll, and if T is conservative, that is, T(c) ⊂ c, then T(c) = c. That is, T restricted to c is an automorphism of c. We had hoped this would hold even if T were not a matrix. We can see, for example, that if the conservative automorphism T is bounded on the unit cube of m by 1 and ρ, where , then T(c) = c. However, in general it is possible for a conservative automorphism of m to map c properly into c.
On the Radius of Curvature for Convex Analytic Functions
Authors: Eenigenburg Paul 
Published: December 2010

Definition 1.1. Let be analytic for |z| < 1. If ƒ is univalent, we say that ƒ belongs to the class S.Definition 1.2. Let ƒ ∈ S, 0 ≦ α < 1. Then ƒ belongs to the class of convex functions of order α, denoted by Kα, provided(1)and if > 0 is given, there exists Z0, |Z0| < 1, such thatLet ƒ ∈ Kα and consider the Jordan curve ϒτ = ƒ(|z| = r), 0 < r < 1. Let s(r, θ) measure the arc length along ϒτ; and let ϕ(r, θ) measure the angle (in the anti-clockwise sense) that the tangent line to ϒτ at ƒ(reiθ) makes with the positive real axis.
A Problem of Gelfand on Rings of Operators and Dynamical Systems
Authors: Kallman Robert R. 
Published: December 2010

Let G be a separable locally compact group (separable in the sense that the topology of G has a countable base). Let S be a standard Borel space on which G acts on the right such that:(1) s · g1g2 = (s · g1) · g2;(2) s · e = s;(3) (s, g) → s · g is a Borel function from S × G to S.If μ is a Borel measure on S, let μg be the Borel measure on S defined by μg(E) = μ(E · g).Let μ be a Borel measure on S which is quasi-invariant under the action of G; i.e., μg and μ are absolutely continuous (g ∈ G). The triple (G, S, μ) is called a dynamical system [11; 8].Consider the following general problem. Let (G, S, μ) be a dynamical system.
On Direct Products of Abelian Groups
Authors: Irwin John M. , O'Neill John D. 
Published: December 2010

In this paper we investigate the properties of the product (or complete direct sum) of torsion Abelian groups. The chief results concern products of Abelian primary groups (p-groups). Given a set of p-groups, [Gλ], over an index set Λ, the product of these groups is written λλ∈ΔGλ, the torsion subgroup of the product of these p-groups is written T[λGλ], and the discrete direct sum of the p-groups is written Σ Gλ.Definition. Σ Gλ is said to be an essentially bounded decomposition if and only if there exists an integer M > 0 such that MGλ = 0 for all but a finite number of Gλs; otherwise the decomposition is essentially unbounded.
On a Conjecture Concerning Semigroup Homomorphisms
Authors: Plemmons R. J. 
Published: December 2010

In this paper we settle (with a counterexample) the question raised by Clifford and Preston in [2, p. 275], concerning maximal group homomorphic images of semigroups. We also consider the question in a more general context and characterize all such examples. The notation and definitions follow [1; 2].By a type of semigroups we mean a class of semigroups, closed under isomorphisms and containing the one-element semigroup. If S is any semigroup and is a type, then a semigroup S* is defined, in [1, p. 18], to be a maximal homomorphic image of S having type if(i) ,(ii) S* is a homomorphic image of S, and(iii) whenever and T is a homomorphic image of S, then there exists a homomorphism from S* onto T.
Screening Properties of the Subbase of all Closed Connected Subsets of a Connectedly Generated Space
Authors: Hursch J. L. 
Published: December 2010

In [1] de Groot has introduced the notation “connectedly generated” (or eg) for those spaces in which the closed connected sets form a subbase for the topology. He pointed out that these are the semi-locally connected spaces of Whyburn. See [5; 6].If X is cg, then, since X is closed, X is the union of a finite number of closed connected sets and, thus, has only a finite number of components. If p is any point in a eg space, and Nv is any neighbourhood of p, then the complement of Nv may be covered by a finite number of closed connected sets, none of which contain p.In [1] and in [2] the concept of “screening” is introduced and shown to be usefully related to local connectedness and construction of compactifications for completely regular spaces. We review this concept in § 2.
Central Idempotent Measures on Unitary Groups
Authors: Rider Daniel 
Published: December 2010

Let G be a locally compact group and M(G) the space of finite regular Borel measures on G. If μ and v are in M(G), their convolution is defined byThus, if f is a continuous bounded function on G,μ is central if μ(Ex) = μ(xE) for all x ∈ G and all measurable sets E. μ is idempotent if μ * μ = μ.The idempotent measures for abelian groups have been classified by Cohen [1]. In this paper we will show that for a certain class of compact groups, containing the unitary groups, the central idempotents can be characterized. The method consists of showing that, in these cases, the central idempotents arise from idempotents on abelian groups and applying Cohen's result.
A Set of Plane Measure Zero Containing all Finite Polygonal Arcs
Authors: Ward D. J. 
Published: December 2010

We say a (plane) set A contains all sets of some type if, for each B of type , there is a subset of A that is congruent to B. Recently, Besicovitch and Rado [3] and independently, Kinney [5] have constructed sets of plane measure zero containing all circles. In these papers it is pointed out that the set of all similar rectangles, some sets of confocal conies and other such classes of sets can be contained in sets of plane measure zero, but all these generalizations rely in some way on the symmetry, or similarity of the sets within the given type.In this paper we construct a set of plane measure zero containing all finite polygonal arcs (i.e., the one-dimensional boundaries of all polygons with a finite number of sides) with slightly stronger results if we restrict our attention to k-gons for some fixed k.
Embedding Theorems for Countable Groups
Authors: McCool James 
Published: December 2010

A group P is said to be a CEF-group if, for every countable group G, there is a factor group of P which contains a subgroup isomorphic to G. It was shown by Higman, Neumann, and Neumann [5] that the free group of rank two is a CEF-group. More recently, Levin [6] proved that if P is the free product of two cyclic groups, not both of order two, then P is a CEF-group. Later, Hall [3] gave an alternative proof of Levin's result.In this paper we give a new proof of Levin's result (Theorem 2). The proof given yields information about the factor group H of P in which a given countable group G is embedded; for example, if G is given by a recursive presentation (this concept is denned in [4]), then a recursive presentation is obtained for H, and certain decision problems (in particular, the word problem) are solvable for the recursive presentation obtained for H if and only if they are solvable for the given recursive presentation of G.