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

Abstract

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 $ϕ:X\rightarrow Y$ into a finite group $Y$ such that $X^ϕ$ is subnormal in $Y$ and $H^ϕ=K\cap X^ϕ$ for a $π$-maximal subgroup $K$ of $Y$. In his talk at the well-known 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.