TY - JOUR

T1 - On groups having the prime graph as alternating and symmetric groups

AU - Gorshkov, Ilya

AU - Staroletov, Alexey

N1 - Publisher Copyright: © 2019, © 2019 Taylor & Francis Group, LLC.

PY - 2019/9/2

Y1 - 2019/9/2

N2 - The prime graph Γ(G) of a finite group G is the graph whose vertex set is the set of prime divisors of |G| and in which two distinct vertices r and s are adjacent if and only if there exists an element of G of order rs. Let A n (S n ) denote the alternating (symmetric) group of degree n. We prove that if G is a finite group with Γ(G)=Γ(A n ) or Γ(G)=Γ(S n ), where n≥19, then there exists a normal subgroup K of G and an integer t such that A t ≤G(K)≤S t and |K| is divisible by at most one prime greater than n/2.

AB - The prime graph Γ(G) of a finite group G is the graph whose vertex set is the set of prime divisors of |G| and in which two distinct vertices r and s are adjacent if and only if there exists an element of G of order rs. Let A n (S n ) denote the alternating (symmetric) group of degree n. We prove that if G is a finite group with Γ(G)=Γ(A n ) or Γ(G)=Γ(S n ), where n≥19, then there exists a normal subgroup K of G and an integer t such that A t ≤G(K)≤S t and |K| is divisible by at most one prime greater than n/2.

KW - alternating groups

KW - finite simple groups

KW - Prime graph

KW - symmetric groups

KW - FINITE

KW - RECOGNITION

KW - RECOGNIZABILITY

UR - http://www.scopus.com/inward/record.url?scp=85063520547&partnerID=8YFLogxK

U2 - 10.1080/00927872.2019.1572167

DO - 10.1080/00927872.2019.1572167

M3 - Article

AN - SCOPUS:85063520547

VL - 47

SP - 3905

EP - 3914

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 9

ER -