Generalized nilpotency in uncountable groups

Abstract

Abstract The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality 𝔪 {\mathfrak{m}} whose proper subgroups of cardinality 𝔪 {\mathfrak{m}} are (bounded) Engel groups. It is proved that such groups are (bounded) Engel groups, provided that they satisfy some generalized solubility condition. A similar analysis is carried out also for (generalized soluble) uncountable groups of cardinality 𝔪 {\mathfrak{m}} whose proper subgroups of cardinality 𝔪 {\mathfrak{m}} are hypercentral. In this case, we get that the whole group is hypercentral provided that the hypercentral lengths of the proper “large” subgroups are not too close to 𝔪 {\mathfrak{m}} . This generalizes results that have already been obtained for nilpotency. Finally, as a by-product, we obtain similar results for many other relevant group classes such as that of Gruenberg groups and that of 𝒩 1 {\mathcal{N}_{1}} -groups.