Abstract
LetG be a finite nonsolvable group andH a proper subgroup ofG. In this paper we determine the structure ofG ifG satisfies one of the following conditions: (1) Every solvable subgroupK(K⊉H) is eitherp-decomposable or a Schmidt group,p being the smallest odd prime factor of |G|. (2) |G∶H| is divisible by an odd prime and every solvable subgroupK(K⊉H) is either 2′-closed or a Schmidt group. (3) |G∶H| is even and every solvable subgroupK(K⊉H) is either 2-closed or a Schmidt group.
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