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Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups. (English) Zbl 07242787
This paper is concerned with algebraic entropy in locally finite groups. The topic of algebraic entropy has been studied in various papers such as [D. Dikranjan et al., Trans. Am. Math. Soc. 361, No. 7, 3401–3434 (2009; Zbl 1176.20057); D. Dikranjan and A. G. Bruno, Topology Appl. 159, No. 13, 2980–2989 (2012; Zbl 1256.54061)]. In this paper the authors discuss whether the addition theorem for algebraic entropy holds for a group $$G$$ in this context. If the additon theorem holds for $$G$$, then it is said that $$AT(G)$$ holds. In this paper the authors are concerned with whether $$AT(G)$$ holds in the case when $$G$$ is locally finite. In this case the algebraic entropy of $$G$$ can be computed via the algebraic entropies of the finite subgroups. Their main result is that for finitely quasihamiltonian locally finite groups $$G$$ it follows that $$AT(G)$$ holds. In particular, if $$G$$ is locally finite and either an FC-group or quasihamiltonian, then $$AT(G)$$ holds.
##### MSC:
 20F65 Geometric group theory 20E07 Subgroup theorems; subgroup growth 20E99 Structure and classification of infinite or finite groups
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