On the permutability of sylow subgroups with Schmidt subgroups

A Schmidt group is a finite nonnilpotent group in which every proper subgroup is nilpotent. Sufficient conditions are established for the p-solvability of a finite group in which a Sylow p-subgroup is permutable with some Schmidt subgroups. Sufficient conditions for the solvability of a finite group in which some Schmidt subgroups are permutable are also obtained.     

On the p-Length of a Product of Two Schmidt Groups

It is established that a finite p-solvable group presenting the product of two of its Schmidt subgroups has p-length at most 2.     

Subgroups of a finite group commuting with biprimary subgroups

The solvability of H/H _G is established under the assumption that a subgroup H of a finite group G commutes with all biprimary subgroups of even order.     

Finite Groups with Subnormal Schmidt Subgroups

We study finite groups some of whose Schmidt subgroups (the minimal nonnilpotent subgroups) are subnormal.     

On the <Emphasis Type="Italic">π</Emphasis>′-properties of a finite group possessing a hall <Emphasis Type="Italic">π</Emphasis>-subgroup

We study the π′-properties of a finite group possessing a Hall π-subgroup that permutes with some Sylow subgroups or some minimal nonnilpotent subgroups.     

Finite groups with seminormal Schmidt subgroups

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. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.