On the Prime Graph of Simple Groups

Abstract

Abstract. Let G be a finite group, let π(G) be the set of prime divisors of |G| and let Γ(G) be the prime graph of G. This graph has vertex set π(G), and two vertices r and s are adjacent if and only if G contains an element of order rs.

Finite Groups with Arithmetic Restrictions on Maximal Subgroups

“…A criterion for coincidence of Gruenberg-Kegel graphs of a simple alternating group and its proper subgroup was obtained modulo the extended binary Goldbach conjecture; for other simple groups, the criterion does not depend on the conjecture. A similar result was independently derived in [16]. For a given set ω of natural numbers, a group G is said to be ω-critical if ω(G) = ω, and for any subgroups K and L of G such that K ¢ L, the equality ω(L/K) = ω implies L = G and K = 1.…”

Equal‐Square Graphs Associated with Finite Groups

“…A large amount of literature is devoted to study the graphs associated to finite groups, for instance, commuting graphs [9][10][11][12][13][14][15], noncommuting graphs [16,17], intersection graphs [18], prime graphs [19][20][21], conjugacy class graphs [22], power graphs [23][24][25], inverse graphs [26], quadratic residues graphs [27], order divisor graphs [28], and square graphs [29].…”

Exponent preserving subgroups of the finite simple groups

“…In the recent years a series of problems have been investigated, related to the existence of a suitable subgroup H ă G preserving some prescribed property of G. For example, Lucchini, Morigi, and Shumyatsky [6] proved that if G is finite then it always contains a 2-generated subgroup H with πpGq " πpHq, and a 3-generated subgroup H with ΓpGq " ΓpHq; Covato [3] extended these results to profinite groups; Burness and Covato [1] showed which finite simple groups G contain a proper subgroup H with ΓpGq " ΓpHq.…”

“…• if e is even, S H and S G are Sylow p-subgroups of GL 2m´1 pqq and GL 2m pqq, respectively. Since |GL 2m pqq| " |GL 2m´1 pqq|¨pq 2m ´1q, they have the same p-power part in the exponent, unless p | q m ´1 and 2m " ep i (see (1)). This can't happen, since p | q m ´1 implies e | m, then 2pm{eq " p i but p is odd;…”

“…If e is even, S H and S G are Sylow p-subgroups of GL 2m´1 pqq and GL 2m`1 pqq, respectively. They may not have the same exponent if 2m `1 " ep i `1 (see (1)), but this can't happen:…”