Showing papers on "Topological group published in 1974"
IBM1
175 citations
Journal Article•
TL;DR: In this paper, a weakly continuous convolution semigroup of positive probability measures on a locally compact Hausdorff topological group is proposed to solve the problem of topological topology.
Abstract: Let G be a locally compact Hausdorff topological group. By a weakly continuous convolution semigroup of positive probability measures on G.
75 citations
01 Jan 1974
TL;DR: It is well known that not every subgroup of a free group is a free topological group as discussed by the authors, however, it is not true in general that every free group subgroup is a group.
Abstract: It is well known that every subgroup of a free group is a free group. However, it is not true in general that a subgroup of a free topological group is a free topological group.
29 citations
27 citations
01 Feb 1974
TL;DR: In this paper, the existence of a group isomorphism between groups C(X, G) and C( Y, G), where X and Y are SQ-pairs, was studied.
Abstract: Let C(X, G) denote the group of continuous functions from a topological space X into a topological group G with the pointwise multiplication. Some classes of SQ-pairs and properties of the corresponding topological group C(X, G) with the compact-open topology are investigated. We also show that the existence of a group isomorphism between groups C(X, G) and C( Y, G) implies the existence of a homeomorphism between X and Y, if (X, G) and (Y, G) are SQ-pairs.
27 citations
27 citations
TL;DR: The set of all combinations, of 3n things taken three at a time, can be partitioned into resolvable Steiner triple systems as discussed by the authors, which can be used to partition the set of combinations.
Abstract: The set of all combinations, of 3n things taken three at a time, can be partitioned into resolvable Steiner triple systems.
26 citations
25 citations
21 citations
TL;DR: In this article, it was shown that if X is locally connected and Hn(X) does not have property L for n > 1, then it is possible to construct peano continua which are nonmovable.
Abstract: Let C denote the category of compact Hausdorff spaces and H: C -* HC be the homotopy functor Let S: C SC be the functor of shape in the sense of Holsztynski for the projection functor H Let X be a continuum and Hn(X) denote n-dimensional Cech cohomology with integer coefficients Let A =char H I(X) be the character group of H 1(X) considering H1(X) as a disx crete group In this paper it is shown that there is a shape morphism F e Morsc(Xs Ax) such that F*: HI(Ax) Ht(X) is an isomorphism It follows from the results of a previous paper by the author that there is a continuous mapping f : X Ax such that S(f =F and thus that f*: H(A )+HI(X) is an isomorphism This result is applied to show that if X is locally connected, then H1(X) has property L Examples are given to show that X may be locally connected and Hn(X) not have property L for n > 1 The result is also applied to compact connected topological groups In the last section of the paper it is shown that if X is compact and movable, then for every integer n, Hn(X)/Tor Hn(X) has property L This result allows us to construct peano continua which are nonmovable An example is given to show that Hn(X) itself may not have property L even if X is a finitedimensional movable continuum Introduction Let C denote the category of compact Hausdorff spaces and H: C -+ HC be the homotopy functor Let S: C - SC be the functor of shape in the sense of HolsztyAski [3] for the projection functor H Let X be a continuum and let Hn(X) denote n-dimensional Cech cohomology with integer coefficients Let Ax be the character group of HW(X) considering H1(X) as a discrete group Note that H1(X) is torsion free and thus Ax is a compact connected abelian topological group In this paper we show that if B is a compact connected abelian topological group and h: H1(B) - H1(X) is a homomorphism, then there is a shape morphism F e Morsc (X, B) such that F* h By the results in [6], there is a continuous function /: x B with S(f) = F and thus with h* Received by the editors November 28, 1972 AMS (MOS) subject classifications (1970) Primary 55D99; Secondary 55BO5, 22B99
20 citations
01 Jan 1974
TL;DR: In this article, it was shown that a necessary and sufficient condition for a free (free abelian) topological group on a topological space X to have no small subgroups is that X admits a continuous metric.
Abstract: The first author has shown that a quotient group of a topological group with no small subgroups can have small subgroups, thus answering a question of Kaplansky in the negative The argument relied on showing that a free abelian topological group on any metric space has no small subgroups Consequently any abelian metric group is a quotient of a group with no small subgroups However metric groups with small subgroups exist in profusion It is shown here that a necessary and sufficient condition for a free (free abelian) topological group on a topological space X to have no small subgroups is that X admits a continuous metric Hence any topological group which admits a continuous metric is a quotient group of a group with no small subgroups
01 Jan 1974
TL;DR: In this article, it was shown that the free product of Hausdorff topological groups G and H is locally compact if and only if G, H and G ╨ H are discrete.
Abstract: In [3], Graev introduced the free product of Hausdorff topological groups G and H (denoted in this paper by G ╨ H) and showed it is algebraically the free product G * H and is Hausdorff. While it has been studied subsequently, for example [4, 6, 7, 8, 11, 12], many questions about its topology remain unsolved. In particular, partial negative results about local compactness were obtained in [7, 11, 12]. In this paper we obtain a complete solution by showing that G ╨ H is locally compact if and only if G, H and G ╨ H are discrete. A similar line of reasoning allows us to show that G ╨ H has no small subgroups if and only if G and H have no small subgroups.
TL;DR: In this article, the question of whether the topological group structure is determined by these weaker structures was considered and it was shown that if G 1 and G 2 are locally compact and connected, then G 1 ≈ G 2 implies G 1 = G 2.
Abstract: An abelian topological group can be considered simply as an abelian group or as a topological space. The question considered in this article is whether the topological group structure is determined by these weaker structures. Denote homeomorphism, isomorphism, and homeomorphic isomorphism by ≈, ≅ , and =, respectively. The principal results are these. Theorem 1. If G 1 and G 2 are locally compact and connected, then G 1 ≈ G 2 implies G 1 = G 2.
TL;DR: For compact topological groups (discrete or continuous) a basis of the group algebra is defined which consists of irreducible tensors only as mentioned in this paper, and this tensor basis is generally discussed and compared with similar constructions for finite groups and SU (2).
Abstract: For compact topological groups (discrete or continuous) a basis of the group algebra is defined which consists of irreducible tensors only. This tensor basis is generally discussed and compared with similar constructions for finite groups and SU (2).
01 Sep 1974
TL;DR: In this paper, the Freudenthal-Weil theorem is used to show that if G and H are compact analytic groups and if G → H is a continuous epimorphism, then o( Z(G) 0 ) = Z(H) 0 where the subscript 0 denotes the identity component of a topological group G and Z (G) its centre.
Abstract: In the proof of the Freudenthal–Weil theorem in, for example (5), essential use is made of the fact that if G and H are compact analytic groups and o: G → H is a continuous epimorphism then o( Z(G) 0 ) = Z(H) 0 where the subscript 0 denotes the identity component of a topological group G and Z(G) its centre. Although this is sufficient for the proof of the Freudenthal–Weil theorem it raises the interesting question as to whether actually o( Z(G) ) = Z(H) (from which the above would follow) and, if so, in what generality this can be expected. The present paper deals with this question, in more general form, as well as certain of its structural consequences.
TL;DR: In this article, the authors use coset diagrams to associate a set of non-negative integers satisfying two conditions with any subgroup of the modular group, which is the specification of the subgroup.
Abstract: With any subgroup of the modular group we associate a set of non-negative integers satisfying two conditions. This set is the specification of the subgroup. Several sets were known which satisfied both conditions, but which did not correspond to subgroups. Here we find several infinite families of such sets. We also investigate situations where a given set corresponds to just one conjugacy class of subgroups. These are related to the well-known lattice subgroups. The method involves the use of coset diagrams, described by Atkin and Swinnerton-Dyer in [1].
TL;DR: The class of almost locally invariant groups is studied in this article, which are groups admitting continuous monomorphisms into local invariants under inner automorphisms, and the class is closed under passage to subgroups, direct products and free products, but not quotients.
Abstract: Maximally almost periodic groups and locally invariant groups have been studied extensively in the literature. Maximally almost periodic groups are those admitting continuous monomorphisms into compact groups; locally invariant groups are those in which every neighbourhood of the identity contains a neighbourhood invariant under inner automorphisms. In this paper a study is made of almost locally invariant groups, which are groups admitting continuous monomorphisms into locally invariant groups. This class includes all maximally almost periodic groups and all locally invariant groups, but there exist locally compact almost locally invariant groups which are neither locally invariant nor maximally almost periodic. A locally compact almost locally invariant group which is connected or locally connected is locally invariant. The class of almost locally invariant groups is closed under passage to subgroups, direct products, and free products, but not quotients.
01 Jan 1974
TL;DR: In this paper, the authors considered compact Hausdorff topological groups, all elements of which have finite order (compact torsion groups), and showed that a compact torsions group is necessarily totally disconnected and hence profinite.
Abstract: This paper is concerned exclusively with compact Hausdorff topological groups, all elements of which have finite order (compact torsion groups). It is well known that a compact torsion group is necessarily totally disconnected [5, (28.20)] and hence profinite.
TL;DR: In this paper, the authors define a topology on an arbitrary lattice-ordered group, called the 5-topology, which is a group and lattice topology for G and which is preserved by cardinal products.
Abstract: Let G be an arbitrary lattice-ordered group. We define a topology on G, called the 5-topology, which is a group and lattice topology for G and which is preserved by cardinal products. The T-topology is the interval topology on totally ordered groups and is discrete if and only if G is a lexico-sum of lexico-extensions of the integers. We derive necessary and sufficient conditions for the 51topology to be Hausdorff, and we investigate 5-topology convergence.
TL;DR: In this paper, it was shown that a compact semigroup with only one-sided continuity (i.e. x↦yx or x ↦xy is continuous for all y ∈ S) is a topological group if and only if the underlying space of S is a compact manifold.
Abstract: It is well known that in a topological semigroup S with an identity 1 the maximal subgroup H (1) must be open if 1 has an euclidean neighborhood (Mostert-Shields [7]). If multiplication in S is only “separately” continuous, i.e. x↦yx and x↦xy is continuous for all y∈S, the statement remains true if the underlying space of S is a compact manifold (Berglund [2], Lawson [6]). In this paper the case of a compact semigroup with only one-sided continuity (i.e. x↦yx or x↦xy is continuous for all y∈S) which is defined on an interval or a circle is investigated. It is also shown that a group, defined on the line or on a circle, must be a topological group if it satisfies this very weak condition.
TL;DR: In this article, it was shown that the universal covering space of a topological group G is the conjugacy space of the action of G on itself by inner automorphisms, and that the Poincare group of G# is locally isomorphic with GIG', the commutator quotient group.
Abstract: The space G# of conjugacy classes of a topological group G is the orbit space of the action of G on itself by inner automorphisms. For a class of connected and locally connected groups which includes all analytic [ZJgroups, the universal covering space of G# may be obtained as the space of conjugacy classes of a group which is locally isomorphic with G, and the Poincare group of G# is found to be isomorphic with that of GIG', the commutator quotient group. In particular, it is shown that the space G# of a compact analytic group G is simply connected if and only if G is semisimple. The proof of this fact has not appeared in the literature, even though more specialized methods are available for this case. I. Defimitions and elementary properties. Two elements x, y of a topological group G are called conjugate, and we write x t y, if there is an element t E G such that y = txt-1. The equivalence class of a point x under this relation is called the conjugacy class of x, denoted Ix. A subset of G which is a union of conjugacy classes is invariant under inner automorphisms and will be said to be invariant. If G acts on itself by inner automorphisms, the inner automorphisms determined by the center Z(G) of G are trivial and G/Z(G) acts effectively on G. The orbit space under the action of G or GIZ(G) is called the space of conjugacy classes of G, denoted G#. If G is the direct product of groups Gi, then G# is homeomorphic with the Cartesian product of the spaces Gf (see [5, p. 130]). The space G# of a compact analytic group G is homeomorphic with the orbit space TIW of the action of the Weyl group W on a maximal toroid T of G [1, p. 95]. If G is semisimple, G# may be obtained by identifying certain boundary points of a compact convex polyhedron in the Lie algebra of T (see [2, Example 6]). Some elementary proofs and [11, p. 231] give the following: LEMmA 1. If G is a compact analytic group, then G# is compact, Hausdorff, second countable, and locally arcwise simply connected. Received by the editors June 1, 1973 and, in revised form, November 5, 1973. AMS (MOS) subject classifications (1970). Primary 57F99; Secondary 54H25, 20F35.
TL;DR: In this article, a kind of topological extensions of a space-time group Q by an electromagnetic gauge group J are investigated in order to determine covariance groups of electrodynamics.
Abstract: A kind of topological extensions of a space‐time group Q by an electromagnetic gauge group J are investigated in order to determine covariance groups of electrodynamics. Here Q stands for the Poincare group, for the Galilei group, or for their neutral components, and J is the Abelian group of all real‐valued functions of class Cm (m ∈ N or m = ∞) defined in space‐time. The topological groups JφfQ so obtained, already important in the study of charged particles in external electromagnetic fields, are analyzed and placed in the general context of combining different symmetry groups. They are characterized by a given operation φ of Q on J and by factor sets f such that f(q,q′) is a constant gauge function for all (q,q′) ∈ Q × Q. It is shown that all these groups JφfQ are topologically isomorphic to the external topological semidirect product of Q by J relative to Φ.
TL;DR: It was shown in this paper that the free k-group on a CW-complex X is itself a CW -complex containing X as a subcomplex, and as a corollary, the free topological group on X is a countable CW complex.
Abstract: It is proved that the free k -group on a CW -complex X is itself a CW -complex containing X as a subcomplex. It follows, as a corollary, that the free topological group on a countable CW -complex is a countable CW -complex.
TL;DR: In this article, it was shown that if X is uniformizable and G is equicontinuous with respect to a compatible uniformity, then G is a topological group.
Abstract: Let X be a topological space and G a subgroup of the homeomorphism group H(X) with the topology of point-wise convergence. It is well-known that if X is uniformizable and G is equicontinuous with respect to a compatible uniformity then G is a topological group. In this paper we show that essentially this same result applies when X is only an R0-space (and hence in particular if X is T 1 or regular). A corresponding result for regular spaces has been proved [2].
TL;DR: In this article, it was shown that if E is the strict inductive limit of a sequence of Mackey spaces {En} such that for every positive integer n, the topological dual space of En, E′n, provided with the Mackey topology μ(E′ n,En), is ultrabornological, then E′[μ(E',E)]/G is complete.
Abstract: In this paper we prove that if E is the strict inductive limit of a sequence of Mackey spaces {En} such that for every positive integer n, the topological dual space of En, E′n, provided with the Mackey topology μ(E′n,En), is ultrabornological, then the topological dual space E′ of E, provided with the Mackey topology μ(E′,E), is ultrabornological. We also prove that if E is a strict (LF)-space and G a closed subspace of E′ [μ(E′,E)] such that E′[μ(E′,E)] /G is sequentially complete, then E′[μ(E′,E)]/G is complete.
TL;DR: A topological group algebra over the vector space is described and a correspondence between inputs and numbers is established, used to prove that the polynomials in the algebra can represent a solution to any pattern recognition problem.
Abstract: Pattern recognition is considered as a mapping from the set of all inputs to the numbers 0 to 1. The inputs are treated as vectors. A topological group algebra over the vector space is described. The input is treated as avariable in a polynomial of that group algebra. A correspondence between inputs and numbers is established. This correspondence is used to prove that the polynomials in the algebra can represent a solution to any pattern recognition problem. When the coefficients of the polynomial are suitably chosen vectors, the natural topology of the input vector space is preserved. The importance of this approach as a basis for a completely general efficient parallel process, and practically realizable pattern recognizing machine is presented. The concept may be realized by a modular parallel process type of machinery.