A Characterization of Finite Simple Groups by the Degrees of Vertices of Their Prime Graphs

Abstract

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if there is an element in G of order pq. Assume [Formula: see text] with primes p1<p2<⋯<pk and natural numbers αi. For p∈π(G), let the degree of p be deg (p)=|{q∈π(G)|q~p}|, and D(G):=( deg (p1), deg (p2),…, deg (pk)). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups Ap with p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC (G)={29.39.5.7, 13}, then G≅S6(3) or O7(3), and finally, we show that if G is a finite group such that |G|=29.39.5.7.13 and D(G)=(3,2,2,1,0), then G≅S6(3) or O7(3).

OD-Characterization of Alternating and Symmetric Groups of Degree 106 and 112

“…we have K = 1 and G ~ A 106 , This complet es the proof of lemma, which concludes the theorem. 13,12,11,10,9,6,5,4,3,1).…”

“…Given a finite group G. denote by Soe( G) the socle of G which is the subgroup generated by the set of all minimal normal In a series of articles (Ref. to [1,3,4,5,6,7,8]), it was shown that many finite almost simple groups arc OD-characterizable, which are included in the following proposition.…”

OD-Characterization of alternating and symmetric groups of degrees 16 and 22

“…[7,10] The simple groups S 6 (3) and O 7 (3) are 2-fold ODcharacterizable, while the automorphism groups Aut(S 6 (3)) and Aut(O 7 (3)) are 6-fold OD-characterizable.…”

“…[7][8][9][10][15][16][17][18] 1) ), it has been shown that many finite almost simple groups are OD-characterizable or k-fold ODcharacterizable for certain k 2. We point out some of these results.…”

“…Result 1 [8][9][10] (a) All sporadic simple groups and their automorphism groups except Aut(J 2 ) and Aut(M c L) are OD-characterizable.…”