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This article is cited in 26 scientific papers (total in 26 papers)
Finite groups with seminormal Schmidt subgroups
V. N. Knyaginaa, V. S. Monakhovb a Gomel Engineering Institute, Ministry of Extraordinary Situations of the Republic of Belarus
b Francisk Skorina Gomel State University
Abstract:
A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt 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 Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt $\{2,3\}$-subgroups and all 5-closed $\{2,5\}$-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary.
Keywords:
finite group, solvable group, Schmidt subgroup, subnormal subgroup, seminormal subgroup.
Received: 14.11.2006 Revised: 23.04.2007
Citation:
V. N. Knyagina, V. S. Monakhov, “Finite groups with seminormal Schmidt subgroups”, Algebra Logika, 46:4 (2007), 448–458; Algebra and Logic, 46:4 (2007), 244–249
Linking options:
https://www.mathnet.ru/eng/al307 https://www.mathnet.ru/eng/al/v46/i4/p448
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