58,140 research outputs found
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 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)
On the uniform domination number of a finite simple group
Let be a finite simple group. By a theorem of Guralnick and Kantor,
contains a conjugacy class such that for each non-identity element , there exists with . Building on this deep
result, we introduce a new invariant , which we call the uniform
domination number of . This is the minimal size of a subset of conjugate
elements such that for each , there exists with . (This invariant is closely related to the total
domination number of the generating graph of , which explains our choice of
terminology.) By the result of Guralnick and Kantor, we have for some conjugacy class of , and the aim of this paper
is to determine close to best possible bounds on for each family
of simple groups. For example, we will prove that there are infinitely many
non-abelian simple groups with . To do this, we develop a
probabilistic approach, based on fixed point ratio estimates. We also establish
a connection to the theory of bases for permutation groups, which allows us to
apply recent results on base sizes for primitive actions of simple groups.Comment: 35 pages; to appear in Trans. Amer. Math. So
Non-existence of integral Hopf orders for twists of several simple groups of Lie type
Let be a prime number and , with if and
otherwise. Let be any non-trivial twist for the complex group
algebra of arising from a -cocycle on an abelian
subgroup of . We show that the twisted Hopf algebra
does not admit a Hopf order over any
number ring. The same conclusion is proved for the Suzuki groups, and for
when the twist stems from an abelian -subgroup. This
supplies new families of complex semisimple (and simple) Hopf algebras that do
not admit a Hopf order over any number ring. The strategy of the proof is
formulated in a general framework that includes the finite simple groups of Lie
type.
As an application, we combine our results with two theorems of Thompson and
Barry and Ward on minimal simple groups to establish that for any finite
non-abelian simple group there is a twist for ,
arising from a -cocycle on an abelian subgroup of , such that
does not admit a Hopf order over any number ring.
This partially answers in the negative a question posed by Meir and the second
author.Comment: 32 pages. Final version. To appear in Publ. Ma
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
Minimal pseudocompact group topologies on free abelian groups
A Hausdorff topological group G is minimal if every continuous isomorphism f:
G --> H between G and a Hausdorff topological group H is open. Significantly
strengthening a 1981 result of Stoyanov, we prove the following theorem: For
every infinite minimal abelian group G there exists a sequence {\sigma_n : n\in
N} of cardinals such that w(G) = sup {\sigma_n : n \in N} and sup {2^{\sigma_n}
: n \in N} \leq |G| \leq 2^{w(G)}, where w(G) is the weight of G. If G is an
infinite minimal abelian group, then either |G| = 2^\sigma for some cardinal
\sigma, or w(G) = min {\sigma: |G| \leq 2^\sigma}; moreover, the equality |G| =
2^{w(G)} holds whenever cf (w(G)) > \omega. For a cardinal \kappa, we denote by
F_\kappa the free abelian group with \kappa many generators. If F_\kappa admits
a pseudocompact group topology, then \kappa \geq c, where c is the cardinality
of the continuum. We show that the existence of a minimal pseudocompact group
topology on F_c is equivalent to the Lusin's Hypothesis 2^{\omega_1} = c. For
\kappa > c, we prove that F_\kappa admits a (zero-dimensional) minimal
pseudocompact group topology if and only if F_\kappa has both a minimal group
topology and a pseudocompact group topology. If \kappa > c, then F_\kappa
admits a connected minimal pseudocompact group topology of weight \sigma if and
only if \kappa = 2^\sigma. Finally, we establish that no infinite torsion-free
abelian group can be equipped with a locally connected minimal group topology.Comment: 18 page
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