Finite groups with seminormal Schmidt subgroups |
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Authors: | V N Knyagina V S Monakhov |
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Institution: | (1) Gomel Engineering Institute under the Belarussian Ministry of Extraordinary Situations, Belarus;(2) Gomel State University, Gomel, Belarus |
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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.
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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Keywords: | finite group solvable group Schmidt subgroup subnormal subgroup seminormal subgroup |
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