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A classification of the prime graphs of pseudo-solvable groups.
- Source :
-
Journal of Group Theory . Jan2024, Vol. 27 Issue 1, p89-117. 29p. - Publication Year :
- 2024
-
Abstract
- The prime graph Γ (G) of a finite group 퐺 (also known as the Gruenberg–Kegel graph) has as its vertices the prime divisors of | G | , and p - q is an edge in Γ (G) if and only if 퐺 has an element of order p q . Since their inception in the 1970s, these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups, such a classification was found in 2015. In this paper, we go beyond solvable groups for the first time and characterize the prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or isomorphic to A 5 . The classification is based on two conditions: the vertices { 2 , 3 , 5 } form a triangle in Γ ̄ (G) or { p , 3 , 5 } form a triangle for some prime p ≠ 2 . The ideas developed in this paper also lay the groundwork for future work on classifying and analyzing prime graphs of more general classes of finite groups. [ABSTRACT FROM AUTHOR]
- Subjects :
- *SOLVABLE groups
*FINITE groups
*TRIANGLES
*CLASSIFICATION
*DIVISOR theory
Subjects
Details
- Language :
- English
- ISSN :
- 14335883
- Volume :
- 27
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Journal of Group Theory
- Publication Type :
- Academic Journal
- Accession number :
- 174661863
- Full Text :
- https://doi.org/10.1515/jgth-2023-0018