61,076 research outputs found
Compact-like abelian groups without non-trivial quasi-convex null sequences
In this paper, we study precompact abelian groups G that contain no sequence
{x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G,
and x_n --> 0. We characterize groups with this property in the following
classes of groups:
(a) bounded precompact abelian groups;
(b) minimal abelian groups;
(c) totally minimal abelian groups;
(d) \omega-bounded abelian groups.
We also provide examples of minimal abelian groups with this property, and
show that there exists a minimal pseudocompact abelian group with the same
property; furthermore, under Martin's Axiom, the group may be chosen to be
countably compact minimal abelian.Comment: Final versio
On non-abelian C-minimal groups
AbstractWe investigate the structure of C-minimal valued groups that are not abelian-by-finite. We prove among other things that they are nilpotent-by-finite
On the quasi-component of pseudocompact abelian groups
In this paper, we describe the relationship between the quasi-component q(G)
of a (perfectly) minimal pseudocompact abelian group G and the quasi-component
q(\widetilde G) of its completion. Specifically, we characterize the pairs
(C,A) of compact connected abelian groups C and subgroups A such that A \cong
q(G) and C \cong q(\widetilde G). As a consequence, we show that for every
positive integer n or n=\omega, there exist plenty of abelian pseudocompact
perfectly minimal n-dimensional groups G such that the quasi-component of G is
not dense in the quasi-component of the completion of G.Comment: minor revisio
The Dynamics of Group Codes: Dual Abelian Group Codes and Systems
Fundamental results concerning the dynamics of abelian group codes
(behaviors) and their duals are developed. Duals of sequence spaces over
locally compact abelian groups may be defined via Pontryagin duality; dual
group codes are orthogonal subgroups of dual sequence spaces. The dual of a
complete code or system is finite, and the dual of a Laurent code or system is
(anti-)Laurent. If C and C^\perp are dual codes, then the state spaces of C act
as the character groups of the state spaces of C^\perp. The controllability
properties of C are the observability properties of C^\perp. In particular, C
is (strongly) controllable if and only if C^\perp is (strongly) observable, and
the controller memory of C is the observer memory of C^\perp. The controller
granules of C act as the character groups of the observer granules of C^\perp.
Examples of minimal observer-form encoder and syndrome-former constructions are
given. Finally, every observer granule of C is an "end-around" controller
granule of C.Comment: 30 pages, 11 figures. To appear in IEEE Trans. Inform. Theory, 200
On automorphism groups of Toeplitz subshifts
In this article we study automorphisms of Toeplitz subshifts. Such groups are
abelian and any finitely generated torsion subgroup is finite and cyclic. When
the complexity is non superlinear, we prove that the automorphism group is,
modulo a finite cyclic group, generated by a unique root of the shift. In the
subquadratic complexity case, we show that the automorphism group modulo the
torsion is generated by the roots of the shift map and that the result of the
non superlinear case is optimal. Namely, for any we construct
examples of minimal Toeplitz subshifts with complexity bounded by whose automorphism groups are not finitely generated. Finally,
we observe the coalescence and the automorphism group give no restriction on
the complexity since we provide a family of coalescent Toeplitz subshifts with
positive entropy such that their automorphism groups are arbitrary finitely
generated infinite abelian groups with cyclic torsion subgroup (eventually
restricted to powers of the shift)
Minimal Faithful Quasi-Permutation Representation Degree of p-Groups with Cyclic Center
For a finite group G, we denote by , and c(G), the minimal degree of
faithful permutation representation of G, and the minimal degree of faithful
representation of G by quasi-permutation matrices over the complex field C,
respectively. In this article, we study , and c(G) for various classes
of finite non-abelian p-groups with cyclic center. We prove a result for
normally monomial p-groups with cyclic center which generalizes a result of
Behravesh for finite p-groups of nilpotency class 2 with cyclic center [5,
Theorem 4.12]. We also compute minimal degrees for some classes of metabelian
p-groups
On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of -groups
For a finite group , we denote by , the minimal degree of faithful
representation of by quasi-permutation matrices over the complex field
. For an irreducible character of , the codegree of
is defined as \cod(\chi) = |G/ \ker(\chi)|/ \chi(1). In this article,
we establish equality between and a -sum of
codegrees of some irreducible characters of a non-abelian -group of odd
order. We also study the relation between and irreducible character
codegrees for various classes of non-abelian -groups, such as, -groups
with cyclic center, maximal class -groups, GVZ -groups, and others
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