TY - JOUR

T1 - On the prime graph of a finite group with unique nonabelian composition factor

AU - Grechkoseeva, Maria A.

AU - Vasil’ev, Andrey V.

N1 - Funding Information: This work was supported by the Program I.1.1 of Fundamental Scientific Researches of the Siberian Branch of Russian Academy of Sciences under Project 0314-2019-0001. Publisher Copyright: © 2022 Taylor & Francis Group, LLC.

PY - 2022

Y1 - 2022

N2 - We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group G isospectral to a finite simple group has a unique nonabelian composition factor, that is, the quotient of G by the solvable radical of G is an almost simple group. The main goal of this paper is to prove that this almost simple group is a cyclic extension of its socle. To this end, we consider a general situation when G is an arbitrary group with unique nonabelian composition factor, not necessarily isospectral to a simple group, and study the prime graph of G, where the prime graph of G is the graph whose vertices are the prime numbers dividing the order of G and two such numbers r and s are adjacent if and only if (Formula presented.) and G has an element of order rs. Namely, we establish some sufficient conditions for the prime graph of such a group to have a vertex adjacent to all other vertices. Besides proving the main result, this allows us to refine a recent result by Cameron and Maslova concerning finite groups almost recognizable by prime graph.

AB - We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group G isospectral to a finite simple group has a unique nonabelian composition factor, that is, the quotient of G by the solvable radical of G is an almost simple group. The main goal of this paper is to prove that this almost simple group is a cyclic extension of its socle. To this end, we consider a general situation when G is an arbitrary group with unique nonabelian composition factor, not necessarily isospectral to a simple group, and study the prime graph of G, where the prime graph of G is the graph whose vertices are the prime numbers dividing the order of G and two such numbers r and s are adjacent if and only if (Formula presented.) and G has an element of order rs. Namely, we establish some sufficient conditions for the prime graph of such a group to have a vertex adjacent to all other vertices. Besides proving the main result, this allows us to refine a recent result by Cameron and Maslova concerning finite groups almost recognizable by prime graph.

KW - Almost simple group

KW - group of Lie type

KW - order of an element

KW - prime graph

KW - recognition by spectrum

UR - http://www.scopus.com/inward/record.url?scp=85125150586&partnerID=8YFLogxK

U2 - 10.1080/00927872.2022.2033254

DO - 10.1080/00927872.2022.2033254

M3 - Article

AN - SCOPUS:85125150586

VL - 50

SP - 3447

EP - 3452

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 8

ER -