## 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 language | English |
---|---|

Pages (from-to) | 3905-3914 |

Number of pages | 10 |

Journal | Communications in Algebra |

Volume | 47 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2 Sep 2019 |

## Keywords

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