On groups having the prime graph as alternating and symmetric groups

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 An (Sn) denote the alternating (symmetric) group of degree n. We prove that if G is a finite group with Γ(G) = Γ(An) or Γ(G) = Γ(Sn), where n ≥ 19, then there exists a normal subgroup K of G and an integer t such that At ≤ G/K ≤ St and |K| is divisible by at most one prime greater than n/2. * The work was supported by the program of fundamental scientific researches of the SB RAS

Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph

“…The question of recognition by Gruenberg-Kegel graph of alternating and symmetric groups was studied in [47,25,76].…”

Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph

“…The question of recognition by Gruenberg-Kegel graph of alternating and symmetric groups was studied in [9,37]. The corresponding question for simple groups S such that |π(S)| ∈ {3, 4} was studied in [24,25].…”

The group J4 × J4 is recognizable by spectrum

“…At the moment, for many finite nonabelian simple groups and their automorphism groups, it was proved that they are recognizable (see, for example, [19]). In 1997 V. D. Mazurov [15,Theorem 2] proved that the direct product of two copies of the group Sz(2 7 ) is recognizable by spectrum. In this paper we prove the following theorem.…”

“…Note that |π(C) ∩ θ| = 6. In view of Lemma 13, there exists a nonabelian composition factor R of C such that 5 ≤ |π(R) ∩ π| ≤ 6 and π(R) ⊆ π(G) \ {p} = {2, 3,5,7,11,23,29,31,37, 43} \ {p}. In view of [21], there is no a finite nonabelian simple group R satisfying these conditions.…”