Uniqueness in Structure Theorems for LCA Groups

1978 ◽  
Vol 30 (03) ◽  
pp. 593-599 ◽  
Author(s):  
D. L. Armacost ◽  
W. L. Armacost
Keyword(s):  
Structure Theorem ◽  
Open Subgroup ◽  
Additive Group ◽  
Real Numbers ◽  
Vector Group ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Usual Topology ◽  
Lca Group ◽  
Lca Groups

The classical Pontrjagin-van Kampen structure theorem states that any locally compact abelian (LCA) group G can be written as the direct product of a vector group Rm (where R denotes the additive group of real numbers with the usual topology, and m is a non-negative integer) and an LCA group H which contains a compact open subgroup. This important theorem, which van Kampen deduced from the work of Pontrjagin, was first stated and proved in [5, p. 461].

scholarly journals Homomorphisms into totally disconnected, locally compact groups with dense image

2019 ◽  
Vol 31 (3) ◽  
pp. 685-701 ◽  
Author(s):  
Colin D. Reid ◽  
Phillip R. Wesolek
Keyword(s):  
Normal Subgroup ◽  
Open Subgroup ◽  
Locally Compact Groups ◽  
Group Homomorphism ◽  
Compact Groups ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Dense Image ◽  
Profinite Completions ◽  
Totally Disconnected

Abstract Let {\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair {(G,\phi^{-1}(L))} , where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.

scholarly journals When are dense subgroups of LCA groups closed under intersections?

1994 ◽  
Vol 49 (1) ◽  
pp. 59-67
Author(s):  
M.A. Khan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Compact Abelian Group ◽  
Additive Group ◽  
Locally Compact Abelian Group ◽  
Torsion Subgroup ◽  
Dense Subgroup ◽  
Locally Compact ◽  
Dense Subgroups ◽  
Lca Groups

Let G be a nondiscrete locally compact Hausdorff abelian group. It is shown that if G contains an open torsion subgroup, then every proper dense subgroup of G is contained in a maximal subgroup; while if G has no open torsion subgroup, then it has a dense subgroup D such that G/D is algebraically isomorphic to R, the additive group of reals. With each G, containing an open torsion subgroup, we associate the least positive integer n such that the nth multiple of every discontinuous character of G is continuous. The following are proved equivalent for a nondiscrete locally compact abelian group G:(1) The intersection of any two dense subgroups of G is dense in G.(2) The intersection of all dense subgroups of G is dense in G.(3) G contains an open torsion subgroup, and for each prime p dividing the positive integer associated with G, pG is either open or a proper dense subgroup of G.Finally, we construct a locally compact abelian group G with infinitely many dense subgroups satisfying the three equivalent conditions stated above.

Locally compact groups with closed subgroups open and p-adic

1995 ◽  
Vol 118 (2) ◽  
pp. 303-313 ◽  
Author(s):  
Karl H. Hofmann ◽  
Sidney A. Morris ◽  
Sheila Oates-Williams ◽  
V. N. Obraztsov
Keyword(s):  
Topological Group ◽  
Open Subgroup ◽  
Additive Group ◽  
Locally Compact Abelian Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Character Group ◽  
Exceptional Groups ◽  
Open Set ◽  
Compact Abelian Groups

An open subgroup U of a topological group G is always closed, since U is the complement of the open set . An arbitrary closed subgroup C of G is almost never open, unless G belongs to a small family of exceptional groups. In fact, if G is a locally compact abelian group in which every non-trivial subgroup is open, then G is the additive group δp of p-adic integers or the additive group Ωp of p-adic rationale (cf. Robertson and Schreiber[5[, proposition 7). The fact that δp has interesting properties as a topological group has many roots. One is that its character group is the Prüfer group ℤp∞, which makes it unique inside the category of compact abelian groups. But even within the bigger class of not necessarily abelian compact groups the p-adic group δp is distinguished: it is the only one all of whose non-trivial subgroups are isomorphic (cf. Morris and Oates-Williams[2[), and it is also the only one all of whose non-trivial closed subgroups have finite index (cf. Morris, Oates-Williams and Thompson [3[).

scholarly journals Totally disconnected locally compact groups locally of finite rank

2015 ◽  
Vol 158 (3) ◽  
pp. 505-530 ◽  
Author(s):  
PHILLIP WESOLEK
Keyword(s):  
Lie Groups ◽  
Finite Rank ◽  
Open Subgroup ◽  
Simple Groups ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Elementary Group ◽  
Finite Set ◽  
Finite Direct Product ◽  
Totally Disconnected

AbstractWe study totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contain a compact open subgroup with finite rank. We show such groups that additionally admit a pro-π compact open subgroup for some finite set of primes π are virtually an extension of a finite direct product of topologically simple groups by an elementary group. This result, in particular, applies to l.c.s.c. p-adic Lie groups. We go on to obtain a decomposition result for all t.d.l.c.s.c. groups containing a compact open subgroup with finite rank. In the course of proving these theorems, we demonstrate independently interesting structure results for t.d.l.c.s.c. groups with a compact open pro-nilpotent subgroup and for topologically simple l.c.s.c. p-adic Lie groups.

Varieties of topological groups III

1970 ◽  
Vol 2 (2) ◽  
pp. 165-178 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Algebraic Variety ◽  
Additive Group ◽  
Topological Groups ◽  
Locally Compact ◽  
Circle Group ◽  
Free Topological Group ◽  
The Family ◽  
Usual Topology

This paper continues the invèstigation of varieties of topological groups. It is shown that the family of all varieties of topological groups with any given underlying algebraic variety is a class and not a set. In fact the family of all β-varieties with any given underlying algebraic variety is a class and not a set. A variety generated by a family of topological groups of bounded cardinal is not a full variety.The varieties V(R) and V(T) generated by the additive group of reals and the circle group respectively each with its usual topology are examined. In particular it is shown that a locally compact Hausdorff abelian group is in V(T) if and only if it is compact. Thus V(R) properly contains V(T).It is proved that any free topological group of a non-indiscrete variety is disconnected. Finally, some comments are made on topologies on free groups.

scholarly journals Cardinalities of locally compact groups and their Stone-Čech compactifications

2003 ◽  
Vol 67 (3) ◽  
pp. 353-364
Author(s):  
Gerald L. Itzkowitz ◽  
Sidney A. Morris ◽  
Vladimir V. Tkachuk
Keyword(s):  
Topological Group ◽  
Discrete Group ◽  
Additive Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Numbers ◽  
Locally Compact ◽  
Euclidean Topology ◽  
Lindelöf Degree

Dedicated to Edwin HewittIf G is any Hausdorff topological group and βG is the Stone-Čech compactification then where |G| denotes the cardinalty of G It is known that if G is a discrete group then and if G is the additive group of real numbers with the Euclidean topology, then |βG| = 2|G|. In this paper the cardinality and weight of βG, for a locally compact group G, is calculated in terms of the character and Lindelöf degree of G The results make it possible to give a reasonably complete description of locally compact groups G for which |βG| = 2|G| or even |βG| = |G|.

Power-Rich and Power-Deficient LCA Groups

1981 ◽  
Vol 33 (3) ◽  
pp. 664-670 ◽  
Author(s):  
M. A. Khan
Keyword(s):  
Haar Measure ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Open Neighbourhood ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Lca Group ◽  
Lca Groups ◽  
Dual Power

In [4], Edwin Hewitt denned a-rich LCA (i.e., locally compact abelian) groups and classified them by their algebraic structure. In this paper, we study LCA groups with some properties related to a-richness. We define an LCA group G to be power-rich if for every open neighbourhood V of the identity in G and for every integer n > 1, λ(nV) > 0, where nV = {nx ∈ G : x ∈ V} and λ is a Haar measure on G. G is power-meagre if for every integer n > 1, there is an open neighbourhood V of the identity, possibly depending on n, such that λ(nV) = 0. G is power-deficient if for every integer n > 1 and for every open neighbourhood V of the identity such that is compact, . G is dual power-rich if both G and Ĝ are power-rich. We define dual power-meagre and dual power-deficient groups similarly.

scholarly journals Density and representation theorems for multipliers of type (p, q)

1967 ◽  
Vol 7 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Alessandro Figà-Talamanca ◽  
G. I. Gaudry
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Important Class ◽  
Lebesgue Spaces ◽  
Representation Theorems ◽  
Locally Compact ◽  
Bounded Linear ◽  
Character Group ◽  
Hausdorff Group ◽  
Lca Group

Let G be a locally compact Abelian Hausdorff group (abbreviated LCA group); let X be its character group and dx, dx be the elements of the normalised Haar measures on G and X respectively. If 1 < p, q < ∞, and Lp(G) and Lq(G) are the usual Lebesgue spaces, of index p and q respectively, with respect to dx, a multiplier of type (p, q) is defined as a bounded linear operator T from Lp(G) to Lq(G) which commutes with translations, i.e. τxT = Tτx for all x ∈ G, where τxf(y) = f(x+y). The space of multipliers of type (p, q) will be denoted by Lqp. Already, much attention has been devoted to this important class of operators (see, for example, [3], [4], [7]).

scholarly journals Uniqueness theorem for Fourier transformable measures on LCA groups

2020 ◽  
Vol 54 (2) ◽  
pp. 211-219
Author(s):  
S.Yu. Favorov
Keyword(s):  
Uniqueness Theorem ◽  
Real Axis ◽  
Almost Periodic ◽  
Periodic Measure ◽  
Locally Compact ◽  
The Real ◽  
Approach Infinity ◽  
The Masses ◽  
Lca Groups

We show that if points of supports of two discrete ”not very thick” Fourier transformable measures on locally compact abelian (LCA) groups tend to one another at infinity and the same is true for the masses at these points, then these measures coincide. The result is valid for discrete almost periodic measures on LCA groups too. Also, we show that the result is false for some discrete ”thick” measures. To do this, we construct a discrete almost periodic measure on the real axis, whose masses at the points of support tend to zero as these points approach infinity.

A Model of the Real Numbers

1963 ◽  
Vol 6 (2) ◽  
pp. 239-255
Author(s):  
Stanton M. Trott
Keyword(s):  
Irrational Number ◽  
Additive Group ◽  
Linear Ordering ◽  
Real Numbers ◽  
Ordered Group ◽  
The Real ◽  
Linear Orderings ◽  
Positive Elements ◽  
The Law ◽  
Ordered Pairs

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.

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