# Classification and properties of the $\pi$-submaximal subgroups in minimal nonsolvable groups

## TL;DR

This paper classifies and analyzes the properties of $oldsymbol{ ext{pi}}$-submaximal subgroups within minimal nonsolvable groups, addressing a longstanding open problem posed by Wielandt regarding their structure and characteristics.

## Contribution

It provides a comprehensive description of $oldsymbol{ ext{pi}}$-submaximal subgroups in minimal nonsolvable groups and explores their properties, solving Wielandt's open problem.

## Key findings

- Classification of $oldsymbol{ ext{pi}}$-submaximal subgroups in minimal nonsolvable groups
- Analysis of properties such as pronormality, intravariancy, and conjugacy
- Resolution of Wielandt's problem on these subgroups

## Abstract

Let $\pi$ be a set of primes. According to H. Wielandt, a subgroup $H$ of a finite group $X$ is called a $\pi$-submaximal subgroup if there is a monomorphism $\phi:X\rightarrow Y$ into a finite group $Y$ such that $X^\phi$ is subnormal in $Y$ and $H^\phi=K\cap X^\phi$ for a $\pi$-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 $\pi$-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 $\pi$ of primes, we obtain a description of the $\pi$-submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt's problem.