Uncountable groups with finitely many normalizers of large subgroups

In this paper, the structure of uncountable groups with finitely many normalizers of large subgroups is studied and the connections between this property and other natural finiteness conditions on large subgroups of uncountable groups are investigated. In particular, groups in which every large subgroup is close to be normal with the only obstruction of a finite section and groups with finitely many commutator subgroups of large subgroups are considered. Moreover, groups with a finite covering consisting of groups with normal large subgroups are studied.