Classification and properties of the π -submaximal subgroups in minimal nonsolvable groups

Let π be a set of primes. According to H. Wielandt, a subgroup H of a finite group X is called a π-submaximal subgroup if there is a monomorphism ϕ: X→ Y into a finite group Y such that X^ϕ is subnormal in Y and H^ϕ= K∩ X^ϕ for a π-maximal subgroup K of Y. In his talk at the celebrated conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the π-submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set π of primes, we obtain a description of the π-submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt’s problem.