58,140 research outputs found

    On non-abelian C-minimal groups

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    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

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    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 ε>0\varepsilon > 0 we construct examples of minimal Toeplitz subshifts with complexity bounded by Cn1+ϵC n^{1+\epsilon} 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

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    Let GG be a finite simple group. By a theorem of Guralnick and Kantor, GG contains a conjugacy class CC such that for each non-identity element x∈Gx \in G, there exists y∈Cy \in C with G=⟨x,y⟩G = \langle x,y\rangle. Building on this deep result, we introduce a new invariant γu(G)\gamma_u(G), which we call the uniform domination number of GG. This is the minimal size of a subset SS of conjugate elements such that for each 1≠x∈G1 \ne x \in G, there exists s∈Ss \in S with G=⟨x,s⟩G = \langle x, s \rangle. (This invariant is closely related to the total domination number of the generating graph of GG, which explains our choice of terminology.) By the result of Guralnick and Kantor, we have γu(G)⩽∣C∣\gamma_u(G) \leqslant |C| for some conjugacy class CC of GG, and the aim of this paper is to determine close to best possible bounds on γu(G)\gamma_u(G) for each family of simple groups. For example, we will prove that there are infinitely many non-abelian simple groups GG with γu(G)=2\gamma_u(G) = 2. 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

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    Let pp be a prime number and q=pmq=p^m, with m≥1m \geq 1 if p≠2,3p \neq 2,3 and m>1m>1 otherwise. Let Ω\Omega be any non-trivial twist for the complex group algebra of PSL2(q)\mathbf{PSL}_2(q) arising from a 22-cocycle on an abelian subgroup of PSL2(q)\mathbf{PSL}_2(q). We show that the twisted Hopf algebra (CPSL2(q))Ω(\mathbb{C} \mathbf{PSL}_2(q))_{\Omega} does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki groups, and for SL3(p)\mathbf{SL}_3(p) when the twist stems from an abelian pp-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 GG there is a twist Ω\Omega for CG\mathbb{C} G, arising from a 22-cocycle on an abelian subgroup of GG, such that (CG)Ω(\mathbb{C} G)_{\Omega} 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

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    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

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    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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