Groups with dense subnormal subgroups
Let \(G\) be a group, \(S\) a family of subgroups. The set \(S\) is said to be dense in \(G\) if for any two subgroups \(H,K\) such that \(H<K\) and \(H\) is not maximal in \(K\) the set \(S\) contains a subgroup \(X\) such that \(H<X<K\). The authors consider groups, in which a family of subnormal subgroups is dense. The main results are the following. Theorem 2.4. Let \(G\) be an infinite group with dense family of subnormal subgroups. Then every subgroup of \(G\) is subnormal. Theorem 3.1. Let \(G\) be an infinite group with dense family of subnormal subgroups of bounded defect. Then \(G\) is nilpotent. Theorem 3.6. Let \(G\) be an infinite group for which the set of subnormal subgroups with defect at most 2 is dense. Then \(G\) is soluble with derived length at most 4.
scientific article; zbMATH DE number 1319589