Maximal and Submaximal x-Subgroups

W. Guo, D. O. Revin

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

Let x be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal x-subgroup if there exists an isomorphic embedding ϕ : G ↪ G * of G into some finite group G * under which G ϕ is subnormal in G * and H ϕ = K ∩ G ϕ for some maximal x-subgroup K of G *. In the case where x coincides with the class of all π-groups for some set π of prime numbers, submaximal x-subgroups are called submaximal π-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal π-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal x- and π-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal x-subgroups are conjugate in a finite group G in which all maximal x-subgroups are conjugate?

Original languageEnglish
Pages (from-to)9-28
Number of pages20
JournalAlgebra and Logic
Volume57
Issue number1
DOIs
Publication statusPublished - 19 May 2018

Keywords

  • Dπ-property
  • finite group
  • Hall π-subgroup
  • maximal 𝔛-subgroup
  • submaximal 𝔛-subgroup
  • maximal x subgroup
  • submaximal x-subgroup
  • D-pi-pproperty
  • maximal x-subgroup
  • CLASSIFICATION
  • Hall pi-subgroup
  • CONJECTURE
  • PRIMITIVE PERMUTATION-GROUPS
  • FINITE-GROUPS
  • HALL SUBGROUPS
  • PI
  • ODD INDEX
  • PRONORMALITY

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