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

Wenbin Guo, Danila O. Revin

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

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→ Y into a finite group Y such that Xϕ is subnormal in Y and Hϕ= K∩ Xϕ 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.

Original languageEnglish
Pages (from-to)325-351
Number of pages27
JournalBulletin of Mathematical Sciences
Volume8
Issue number2
DOIs
Publication statusPublished - 1 Aug 2018

Keywords

  • Minimal nonsolvable group
  • Minimal simple group
  • Pronormal subgroup
  • π-Maximal subgroup
  • π-Submaximal subgroup
  • FINITE SIMPLE-GROUPS
  • EXISTENCE
  • pi-Maximal subgroup
  • THEOREM
  • CONJECTURE
  • SYLOW TYPE
  • HALL SUBGROUPS
  • PRONORMALITY
  • CRITERION
  • pi-Submaximal subgroup

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