Displaying similar documents to “Cellularity of a space of subgroups of a discrete group”

Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan, Dmitri Shakhmatov (2013)

Fundamenta Mathematicae

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According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or G δ -dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined...

On normal subgroups of compact groups

Nikolay Nikolov, Dan Segal (2014)

Journal of the European Mathematical Society

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Among compact Hausdorff groups G whose maximal profinite quotient is finitely generated, we characterize those that possess a proper dense normal subgroup. We also prove that the abstract commutator subgroup [ H , G ] is closed for every closed normal subgroup H of G .

Isolated subgroups of finite abelian groups

Marius Tărnăuceanu (2022)

Czechoslovak Mathematical Journal

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We say that a subgroup H is isolated in a group G if for every x ∈ G we have either x ∈ H or 〈 x 〉 ∩ H = 1 . We describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.

Connected transversals -- the Zassenhaus case

Tomáš Kepka, Petr Němec (2000)

Commentationes Mathematicae Universitatis Carolinae

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In this short note, it is shown that if A , B are H -connected transversals for a finite subgroup H of an infinite group G such that the index of H in G is at least 3 and H ∩ H u ∩ H v = 1 whenever u , v ∈ G ∖ H and u v - 1 ∈ G ∖ H then A = B is a normal abelian subgroup of G .

Ulm-Kaplansky invariants of S(KG)/G

P. V. Danchev (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let G be an infinite abelian p-group and let K be a field of the first kind with respect to p of characteristic different from p such that s p ( K ) = ℕ or s p ( K ) = ℕ ∪ 0 . The main result of the paper is the computation of the Ulm-Kaplansky functions of the factor group S(KG)/G of the normalized Sylow p-subgroup S(KG) in the group ring KG modulo G. We also characterize the basic subgroups of S(KG)/G by proving that they are isomorphic to S(KB)/B, where B is a basic subgroup of G.

The generalized criterion of Dieudonné for valuated p -groups

Peter Vassilev Danchev (2006)

Acta Mathematica Universitatis Ostraviensis

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We prove that if G is an abelian p -group with a nice subgroup A so that G / A is a Σ -group, then G is a Σ -group if and only if A is a Σ -subgroup in G provided that A is equipped with a valuation induced by the restricted height function on G . In particular, if in addition A is pure in G , G is a Σ -group precisely when A is a Σ -group. This extends the classical Dieudonné criterion (Portugal. Math., 1952) as well as it supplies our recent results in (Arch. Math. Brno, 2005), (Bull. Math. Soc....

Notes on the average number of Sylow subgroups of finite groups

Jiakuan Lu, Wei Meng, Alexander Moretó, Kaisun Wu (2021)

Czechoslovak Mathematical Journal

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We show that if the average number of (nonnormal) Sylow subgroups of a finite group is less than 29 4 then G is solvable or G / F ( G ) ≅ A 5 . This generalizes an earlier result by the third author.