Finite groups with seminormal Schmidt subgroups. (Russian, English) Zbl 1155.20016
Algebra Logika 46, No. 4, 448-458 (2007); translation in Algebra Logic 46, No. 4, 244-249 (2007).
Summary: A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Shmidt group. A subgroup \(A\) is said to be seminormal in a group \(G\) if there exists a subgroup \(B\) such that \(G=AB\) and \(AB_1\) is a proper subgroup of \(G\), for every proper subgroup \(B_1\) of \(B\).
Groups that contain seminormal Shmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Shmidt \(\{2,3\}\)-subgroups and all \(5\)-closed \(\{2,5\}\)-Shmidt subgroups of the group are seminormal; the classification of finite groups is not used in doing so. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary.
Groups that contain seminormal Shmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Shmidt \(\{2,3\}\)-subgroups and all \(5\)-closed \(\{2,5\}\)-Shmidt subgroups of the group are seminormal; the classification of finite groups is not used in doing so. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary.
MSC:
20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
20D15 | Finite nilpotent groups, \(p\)-groups |
20D40 | Products of subgroups of abstract finite groups |
20D35 | Subnormal subgroups of abstract finite groups |