A subgroup X of a group G is closed in the profinite topology if it can be obtained as intersection of a collection of subgroups of finite index of G, and a group G is said to be an ERF-group if all its subgroups are closed. It is proved here that if all large subgroups of an uncountable group G are closed, then G is an ERF-group, provided that either G is nilpotent-by-finite or it has finite conjugacy classes. Moreover, uncountable groups in which every large proper subgroup is an ERF-group are considered.
Uncountable extended residually finite groups / De Falco, M.; Musella, C.; Zaccardo, A.. - In: RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO. - ISSN 1973-4409. - 74:4(2025). [10.1007/s12215-025-01221-9]
Uncountable extended residually finite groups
De Falco M.;Musella C.;Zaccardo A.
2025
Abstract
A subgroup X of a group G is closed in the profinite topology if it can be obtained as intersection of a collection of subgroups of finite index of G, and a group G is said to be an ERF-group if all its subgroups are closed. It is proved here that if all large subgroups of an uncountable group G are closed, then G is an ERF-group, provided that either G is nilpotent-by-finite or it has finite conjugacy classes. Moreover, uncountable groups in which every large proper subgroup is an ERF-group are considered.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
