Locally finite groups and configurations

Let 𝐺 be a discrete group. In 2001, Rosenblatt and Willis proved that 𝐺 is amenable if and only if every possible system of configuration equations admits a normalized solution. In this paper, we show independently that 𝐺 is locally finite if and only if every possible system of configuration equations admits a strictly positive solution. Also, we give a procedure to get equidecomposable subsets 𝐴 and 𝐵 of an infinite finitely generated or a locally finite group 𝐺 such that A ⊊ B A\subsetneq B , directly from a system of configuration equations not having a strictly positive solution.

On infinite groups in which all abelian subgroups are locally cyclic

The structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups

Additivity with respect to exact sequences is, notoriously, a fundamental property of the algebraic entropy of group endomorphisms. It was proved for abelian groups by using the structure theorems for such groups in an essential way. On the other hand, a solvable counterexample was recently found, showing that it does not hold in general. Nevertheless, we give a rather short proof of the additivity of algebraic entropy for locally finite groups that are either quasihamiltonian or FC-groups.

The structure of locally finite groups of finite c-dimension

The [Formula: see text]-dimension of a group is the supremum of lengths of strict nested chains of centralizers. We describe the structure of locally finite groups of finite [Formula: see text]-dimension. We also prove that the [Formula: see text]-dimension of the quotient [Formula: see text] of a locally finite group [Formula: see text] by the locally soluble radical [Formula: see text] is bounded in terms of the [Formula: see text]-dimension of [Formula: see text].

Locally finite groups of finite centralizer dimension

We describe the structure of locally finite groups of finite centralizer dimension.