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Addendum to : “Prime graph components of finite almost simple groups” [Rend. Sem. Mat. Univ. Padova 102 (1999), 1-22]

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REND. SEM. MAT. UNIV. PADOVA, Vol. 107 (2002)

Addendum to «Prime graph components of finite almost simple groups».

MARIA SILVIA LUCIDO(*)

This is an addendum to the paper [2]. We refer to that paper for the definitions. We denote by G(G) the prime graph of a finite group G.

The following groups have to be added to the list of the almost simple groups in which the prime graph is not connected in [2].

PROPOSITION 1. Let G be one of the following almost simple groups:

SEGGAut (S) and G does not split over S, with S a simple non abelian group such that G(S) is not connected;

PGL( 2 , q), with q4pn, p an odd prime;

G4Sz(q)aabwhere q42f,f is an odd prime number andais a field automorphism of order f;

G4Ree(q)aabwhere q43f,f is an odd prime number andais a field automorphism of order f.

PSL( 3 , 4 )4SEGEAut (S) and NG/SN42.

Then G(G) is not connected.

PROOF. LetGbe an almost simple groupSEGGAut (S) such thatG does not split, we prove that G(G)4G(S). As at the beginning of the proof of the case of Lie groups in [2], we can suppose thatNG/SN 4p. Let xbe an element of G0S, thenxpS butxpc1. Moreover we can sup- pose thatxis ap-element. ThenCS(x)GCS(xp), therefore for any prime rp(G) we have that rAp in G(G) if and only if rAp in G(S).

IfG4PGL( 2 ,q) withq4pf,pan odd prime, thenp1(G)4p(q221 ) and p2(G)4 ]p(.

Let G4Sz(q)aab whereq42f,f is an odd prime number and a is a field automorphism of orderf. IfS4Sz(q)42B2(q), then the connected

(*) Indirizzo dell’A.: Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienze 200, I-33100 Udine, Italy; e-mail: lucidoHdimi.uniud.it

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Maria Silvia Lucido 190

components of G(S) not containing 2 are

p2(S)4p(q21 ), p3(S)4p(q2k2q11 ), p4(S)4p(q1k2q11 ) . Ifais a field automorphism, thenp(CS(a) )4p(N2B2( 2 )N)4 ]2 , 5( (see [1], Theorem 9.1). We observe that 5 divides 22f11. It can be proved that, ifff1 , 7 (8), 5 divides (q1k2q11 ); or, ifff3 , 5 (8), then 5 di- vides (q2k2q11 ). Therefore

p1(G)4]2,f(Np4(S) p2(G)4p2(S) p3(G)4p3(S) if ff1, 7 (8), p1(G)4]2,f(Np3(S) p2(G)4p2(S) p3(G)4p4(S) if ff3, 5 (8).

LetG4Ree(q)aabwhereq43f,fis an odd prime number and gis a field automorphism of orderf. IfS4Ree(q)42G2(q2), then the connect- ed components of G(S) not containing 2 are

p2(S)4p(q2k3q11 ), p3(S)4p(q1k3q11 ) .

If a is a field automorphism, then p(CS(a) )4p(N2G2( 3 )N)4 ]2 , 3 , 7( (see [1], Theorem 9.1). We observe that 7 divides 33f11. It can be proved that, ifff1 , 11 (12), 7 divides (q1k3q11 ); or, if ff5 , 7 (12), then 7 divides (q2k3q11 ). Therefore

p1(G)4p(q(q221 ) f)Np3(S) p2(G)4p2(S) if ff1 , 11 (12), p1(G)4p(q(q221 ) f)Np2(S) p2(G)4p3(S) if ff5 , 7 (12).

IfS4PSL( 3 , 4 ) and GGAut (S) with NG/SN 42 , then we have the following three possibilities:

G4S.21, extension with a graph-field automorphism, then p1(G)4 4 ]2 , 3(, p2(G)4 ]5( and p3(G)4 ]7(.

G4S.22, extension with a field automorphism, then p1(G)4 4 ]2 , 3 , 7(, p2(G)4 ]5(.

G4S.23, extension with a graph automorphism, then p1(G)4 4 ]2 , 3 , 5(, p2(G)4 ]7(.

R E F E R E N C E S

[1] D. GORENSTEIN- R. LYONS,The local structure of finite groups of characteris- tic 2 type, Memoirs of the American Mathematical Society276, March 1983 (vol. 42).

[2] M. S. LUCIDO, Prime Graph Components of Finite Almost Simple Groups, Rendiconti Sem. Mat. Università di Padova, 102(1999), pp. 1-22.

Manoscritto pervenuto in redazione il 19 ottobre 2001.

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