A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of a group G is permutably embedded in G if is quasihamiltonian for each element g of G. It is proved here that if a group G contains a permutably embedded normal subgroup Q such G/Q that is Cernikov, then G has a quasihamiltonian subgroup of finite index; moreover, if G is periodic, then it contains a Cernikov normal subgroup N such that G/N is quasihamiltonian. This result should be compared with theorems of Cernikov and Schlette stating that if a group G is Cernikov over its centre, then G is abelian-by-finite and its commutator subgroup is Cernikov.
GROUPS with A LARGE PERMUTABLY EMBEDDED SUBGROUP / De Falco, M.; De Giovanni, F.; Musella, C.. - In: BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY. - ISSN 0004-9727. - 107:2(2023), pp. 276-283. [10.1017/S0004972722000648]
GROUPS with A LARGE PERMUTABLY EMBEDDED SUBGROUP
De Falco M.;De Giovanni F.
;Musella C.
2023
Abstract
A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of a group G is permutably embedded in G if is quasihamiltonian for each element g of G. It is proved here that if a group G contains a permutably embedded normal subgroup Q such G/Q that is Cernikov, then G has a quasihamiltonian subgroup of finite index; moreover, if G is periodic, then it contains a Cernikov normal subgroup N such that G/N is quasihamiltonian. This result should be compared with theorems of Cernikov and Schlette stating that if a group G is Cernikov over its centre, then G is abelian-by-finite and its commutator subgroup is Cernikov.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
