2005
DOI: 10.1007/s10469-005-0037-5
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An Adjacency Criterion for the Prime Graph of a Finite Simple Group

Abstract: UDC 512.542Keywords: finite group, finite simple group, group of Lie type, spectrum of a finite group, recognition by spectrum, prime graph of a finite group, independence number of a prime graph, 2-independence number of a prime graph.For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertic… Show more

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Cited by 149 publications
(280 citation statements)
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“…Also since 2n < 2(2n − 2), we have sβ = (2n − 2)α and s < t. Now we consider each possibility for S, separately. If ρ(p, S) = {r i | i ∈ I} ∪ {p}, then using the results in [19], each r j ∈ π(S), where j ∈ I, is adjacent to p in Γ(S). Consequently, we get that β = 2α and m = n, that is S ∼ = L n (p 2α ).…”
Section: Lemma 32 If S Is Isomorphic To a Classical Simple Group Ofmentioning
confidence: 99%
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“…Also since 2n < 2(2n − 2), we have sβ = (2n − 2)α and s < t. Now we consider each possibility for S, separately. If ρ(p, S) = {r i | i ∈ I} ∪ {p}, then using the results in [19], each r j ∈ π(S), where j ∈ I, is adjacent to p in Γ(S). Consequently, we get that β = 2α and m = n, that is S ∼ = L n (p 2α ).…”
Section: Lemma 32 If S Is Isomorphic To a Classical Simple Group Ofmentioning
confidence: 99%
“…Let S ∼ = B m (q ) or C m (q ). Since t(p, S) ≥ 3, using [19,Tables 4], we get that m is odd. In this case ρ(p, S) = {p, r m , r 2m }.…”
Section: Lemma 32 If S Is Isomorphic To a Classical Simple Group Ofmentioning
confidence: 99%
“…If S is a sporadic simple group or an exceptional simple group of Lie type, then t(S) ≤ 12 (see Table 4 in [11] and Table 2 in [10]). But this is impossible, because by Lemma (2.1)(2), t(S) ≥ t(G)−1 ≥ 13.…”
Section: Proof Of the Main Theoremmentioning
confidence: 99%
“…Now, we are ready to prove the main results. [1,[7][8][9][10], S is isomorphic to one of the groups A 2 (q) (q + 1 = 2 k and (q − 1)…”
Section: Proof Of the Theoremsmentioning
confidence: 99%
“…Following [9], we introduce the following notation: if q is a positive integer, r is an odd prime, and (r, q) = 1, then e(r, q) is the minimal positive integer n with the condition q n ≡ 1 (mod r). If r = 2, then we set e(2, q) = 1 for q ≡ 1 (mod 4) and e(2, q) = 2 for q ≡ 3 (mod 4).…”
Section: Proof Of Theorem 1 By Statement (A) Of Lemma 3 We Have T(gmentioning
confidence: 99%