On some subgroup chains related to Kneser’s theorem

Yahya Ould Hamidoune; Oriol Serra; Gilles Zémor

Displaying similar documents to “On some subgroup chains related to Kneser’s theorem”

Properties of subgroups not containing their centralizers

Lemnouar Noui (2009)

Annales mathématiques Blaise Pascal

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In this paper, we give a generalization of Baer Theorem on the injective property of divisible abelian groups. As consequences of the obtained result we find a sufficient condition for a group $G$ to express as semi-direct product of a divisible subgroup $D$ and some subgroup $H$ . We also apply the main Theorem to the $p$ -groups with center of index $p^{2}$ , for some prime $p$ . For these groups we compute $N_{c} (G)$ the number of conjugacy classes and $N_{a}$ the number of abelian maximal subgroups and $N_{n a}$ the number...

Representation of finite abelian group elements by subsequence sums

David J. Grynkiewicz, Luz E. Marchan, Oscar Ordaz (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $G ≅ C_{n_{1}} \oplus ... \oplus C_{n_{r}}$ be a finite and nontrivial abelian group with $n_{1} | n_{2} | ... | n_{r}$ . A conjecture of Hamidoune says that if $W = w_{1} \cdot ... \cdot w_{n}$ is a sequence of integers, all but at most one relatively prime to $| G |$ , and $S$ is a sequence over $G$ with $| S | \geq | W | + | G | - 1 \geq | G | + 1$ , the maximum multiplicity of $S$ at most $| W |$ , and $σ (W) \equiv 0 mod | G |$ , then there exists a nontrivial subgroup $H$ such that every element $g \in H$ can be represented as a weighted subsequence sum of the form $g = \underset{i = 1}{\sum^{n}} w_{i} s_{i}$ , with $s_{1} \cdot ... \cdot s_{n}$ a subsequence of $S$ . We give two examples showing this does not hold in general, and characterize the...

Characterizations of groups generated by Kronecker sets

András Biró (2007)

Journal de Théorie des Nombres de Bordeaux

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In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus $T = R / Z$ by subsets of $Z$ . Here we consider new types of subgroups: let $K \subseteq T$ be a Kronecker set (a compact set on which every continuous function $f : K \to T$ can be uniformly approximated by characters of $T$ ), and $G$ the group generated by $K$ . We prove (Theorem 1) that $G$ can be characterized by a subset of $Z^{2}$ (instead of a subset of $Z$ ). If $K$ is finite, Theorem 1 implies our...

Small-sum pairs in abelian groups

Reza Akhtar, Paul Larson (2010)

Journal de Théorie des Nombres de Bordeaux

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Let $G$ be an abelian group and $A, B$ two subsets of equal size $k$ such that $A + B$ and $A + A$ both have size $2 k - 1$ . Answering a question of Bihani and Jin, we prove that if $A + B$ is aperiodic or if there exist elements $a \in A$ and $b \in B$ such that $a + b$ has a unique expression as an element of $A + B$ and $a + a$ has a unique expression as an element of $A + A$ , then $A$ is a translate of $B$ . We also give an explicit description of the various counterexamples which arise when neither condition holds.

Wintenberger’s functor for abelian extensions

Kevin Keating (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$ -adic Lie extensions $E / F$ , where $F$ is a local field with residue field $k$ , and a category whose objects are pairs $(K, A)$ , where $K ≅ k ((T))$ and $A$ is an abelian $p$ -adic Lie subgroup of ${Aut}_{k} (K)$ . In this paper we extend this equivalence to allow $Gal (E / F)$ and $A$ to be arbitrary abelian pro- $p$ groups.