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 AB1 is a proper subgroup of G, for every proper subgroup B1 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.
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Supported by BelFBR grant Nos. F05-341 and F06MS-017.
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Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.
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Knyagina, V.N., Monakhov, V.S. Finite groups with seminormal Schmidt subgroups. Algebr Logic 46, 244–249 (2007). https://doi.org/10.1007/s10469-007-0023-1
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DOI: https://doi.org/10.1007/s10469-007-0023-1