Cocliques of maximal size in the prime graph of a finite simple group

Abstract

A prime graph of a finite group is defined in the following way: the set of vertices of the graph is the set of prime divisors of the group, and two distinct vertices r and s are adjacent, if there is an element of order rs in the group. In this paper we continue our investigation of the prime graph of a finite simple group started in [1], namely we describe all cocliques of maximal size for all finite simple groups.

Quasirecognition by prime graph of $C_{n}(4) $, where $ n\geq17 $ is odd

“…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.…”

On finite groups with disconnected prime graph

“…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)…”

“…Moreover, 3 and 5 are the only nonadjacent vertices in π 1 (G). By [1,[7][8][9][10], S is isomorphic to one of the groups A 2 (q) (q +1 = 2 k and (q − 1) 3 = 3), A 2 (q) (q +1 = 2 k and (q − 1) 3 = 3) for odd q, 2 A 2 (q) (q − 1 = 2 k and (q + 1) 3 = 3), 2 A 2 (q) (q − 1 = 2 k and (q + 1) 3 = 3) for odd q, A 3 (3), A 4 (q) for odd q, A 5 (2),…”

“…Note that the numbers 3 and 5 are vertices from the intersection of maximal cocliques; hence, in view of [10], S is not isomorphic to the following groups: D 4 (3), He, McL, and HN . Let us use [10] to investigate in detail vertices of the intersection of maximal cocliques for other groups from the list in the preceding paragraph.…”

“…Let us use [10] to investigate in detail vertices of the intersection of maximal cocliques for other groups from the list in the preceding paragraph.…”