A group is called a V-group if it has finite conjugacy classes of subnormal subgroups. It is proved here that if G is a periodic soluble group in which every subnormal subgroup of infinite rank has finitely many conjugates, then G is a V-group, provided that its Hirsch–Plotkin radical has infinite rank. Corresponding results for periodic soluble groups in which every subnormal subgroup of infinite rank has finite index in its normal closure and for those in which every subnormal subgroup of infinite rank is finite over its core, are also obtained. Moreover, it is shown that the assumption on the Hirsch–Plotkin radical can be avoided in the case of periodic groups with nilpotent commutator subgroup.
Groups of infinite rank with finite conjugacy classes of subnormal subgroups / DE FALCO, Maria; DE GIOVANNI, Francesco; Musella, Carmela; Trabelsi, N.. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 431:(2015), pp. 24-37. [10.1016/j.jalgebra.2015.01.031]
Groups of infinite rank with finite conjugacy classes of subnormal subgroups
DE FALCO, MARIA;DE GIOVANNI, FRANCESCO;MUSELLA, CARMELA;
2015
Abstract
A group is called a V-group if it has finite conjugacy classes of subnormal subgroups. It is proved here that if G is a periodic soluble group in which every subnormal subgroup of infinite rank has finitely many conjugates, then G is a V-group, provided that its Hirsch–Plotkin radical has infinite rank. Corresponding results for periodic soluble groups in which every subnormal subgroup of infinite rank has finite index in its normal closure and for those in which every subnormal subgroup of infinite rank is finite over its core, are also obtained. Moreover, it is shown that the assumption on the Hirsch–Plotkin radical can be avoided in the case of periodic groups with nilpotent commutator subgroup.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.