On groups having the prime graph as alternating and symmetric groups

Ilya Gorshkov, Alexey Staroletov

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)3905-3914
Number of pages10
JournalCommunications in Algebra
Volume47
Issue number9
DOIs
Publication statusPublished - 2 Sep 2019

Keywords

  • alternating groups
  • finite simple groups
  • Prime graph
  • symmetric groups
  • FINITE
  • RECOGNITION
  • RECOGNIZABILITY

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