Permutability in uncountable groups
A group is \textit{quasihamiltonian} if each of its subgroups is permutable (or quasinormal), which means that \(HK=KH\) for all subgroups \(H\) and \(K\) of the group. Permutable subgroups were introduced by \textit{Ø. Ore} [Duke Math. J. 3, 149--174 (1937; Zbl 0016.35103)] and \textit{K. Iwasawa} [Jpn. J. Math. 18, 709--728 (1943; Zbl 0061.02504)] completely described quasihamiltonian groups in 1943. Recently, groups whose uncountable subgroups are normal have been studied (see [\textit{M. de Falco} et al., Adv. Group Theory Appl. 3, 13--29 (2017; Zbl 1390.20031)]) and the goal of this work is to do a similar investigation using permutable subgroups. Examples of \textit{A. Ehrenfeucht} and \textit{V. Faber} [Nederl. Akad. Wet., Proc., Ser. A 75, 202--209 (1972; Zbl 0256.20032)] show that uncountable groups whose uncountable subgroups are permutable are not always quasihamiltonian groups, even in the periodic nilpotent case, and the authors point out that the absence of large (i.e. uncountable) abelian subgroups is decise in this context (Lemma 2.3). In any case, the authors are able to prove that uncountable groups whose large subgroups are permutable have a proper commutator subgroup whenever there are no large homomorphic images (Theorem 2.10) and hence a small (i.e. countable) second commutator subgroup (Corollary 2.11). The study splits in two cases according to which the group is periodic or not, but in both cases under the assumption that there are no large simple sections. In the non-periodic case they manage to obtain a complete characterization (Theorem 3.5) that is similar to that of Iwasawa (see [\textit{R. Schmidt}, Subgroup lattices of groups. Berlin: Walter de Gruyter (1994; Zbl 0843.20003)], Theorem 2.4.11). In the periodic case, they show that such groups contain a large direct factor which is a locally finite \(p\)-group (Theorem 4.1).
scientific article; zbMATH DE number 7001285
scientific article