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Group Extensions with Infinite Conjugacy Classes

Jean-Philippe Préaux (2013)

Confluentes Mathematici

We characterize the group property of being with infinite conjugacy classes (or icc, i.e. infinite and of which all conjugacy classes except { 1 } are infinite) for groups which are extensions of groups. We prove a general result for extensions of groups, then deduce characterizations in semi-direct products, wreath products, finite extensions, among others examples we also deduce a characterization for amalgamated products and HNN extensions. The icc property is correlated to the Theory of von Neumann...

Groups of given intermediate word growth

Laurent Bartholdi, Anna Erschler (2014)

Annales de l’institut Fourier

We show that there exists a finitely generated group of growth ∼ f for all functions f : ℝ + → ℝ + satisfying f ( 2 R ) ≤ f ( R ) 2 ≤ f ( η + R ) for all R large enough and η + ≈ 2 . 4675 the positive root of X 3 - X 2 - 2 X - 4 . Set α - = log 2 / log η + ≈ 0 . 7674 ; then all functions that grow uniformly faster than exp ( R α - ) are realizable as the growth of a group.We also give a family of sum-contracting branched groups of growth ∼ exp ( R α ) for a dense set of α ∈ [ α - , 1 ] .

Groups with Decomposable Set of Quasinormal Subgroups

de Falco, M., de Giovanni, F. (2001)

Serdica Mathematical Journal

A subgroup H of a group G is said to be quasinormal if HX = XH for all subgroups X of G. In this article groups are characterized for which the partially ordered set of quasinormal subgroups is decomposable.

Groups with every subgroup ascendant-by-finite

Sergio Camp-Mora (2013)

Open Mathematics

A subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.

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