On the ? -Length of Maximal Subgroups of Finite ? -Soluble Groups

Abd El-Rahman Heliel, Mohammed Al-Shomrani and Adolfo Ballester-Bolinches
Additional contact information
Abd El-Rahman Heliel: Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Mohammed Al-Shomrani: Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Adolfo Ballester-Bolinches: Departament de Matemàtiques, Universitat de València, Dr. Moliner 50, Burjassot, 46100 València, Spain

Mathematics, 2020, vol. 8, issue 12, 1-4

Abstract: Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ -primary if all the prime factors of | G | belong to the same member of σ . G is said to be σ - soluble if every chief factor of G is σ -primary, and G is σ -nilpotent if it is a direct product of σ -primary groups. It is known that G has a largest normal σ -nilpotent subgroup which is denoted by F σ ( G ) . Let n be a non-negative integer. The n -term of the σ -Fitting series of G is defined inductively by F 0 ( G ) = 1 , and F n + 1 ( G ) / F n ( G ) = F σ ( G / F n ( G ) ) . If G is σ -soluble, there exists a smallest n such that F n ( G ) = G . This number n is called the σ - nilpotent length of G and it is denoted by l σ ( G ) . If F is a subgroup-closed saturated formation, we define the σ - F - length n σ ( G , F ) of G as the σ -nilpotent length of the F -residual G F of G . The main result of the paper shows that if A is a maximal subgroup of G and G is a σ -soluble, then n σ ( A , F ) = n σ ( G , F ) − i for some i ∈ { 0 , 1 , 2 } .

Keywords: finite group; ? -solubility; ? -nilpotency; ? -nilpotent length (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/12/2165/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/12/2165/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:12:p:2165-:d:457252

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().