A note on commutator subgroups in groups of large cardinality

Abstract

If G is an uncountable group of regular cardinality $$\aleph $$, we shall denote by $${\mathfrak {L}L}_\aleph (G)$$ the set of all subgroups of G of cardinality $$\aleph $$. The aim of this paper is to describe the behaviour of groups G for which the set $${{\mathcal {C}}}_\aleph (G)=\{ X'\;|\; X\in {\mathfrak {L}L}_\aleph (G)\}$$ is finite, at least when G is locally graded and has no simple sections of cardinality $$\aleph $$. Among other results, it is proved that such a group has a finite commutator subgroup, provided that it contains an abelian subgroup of cardinality $$\aleph $$.