On recognition of simple group L 2 ( r ) by the number of Sylow subgroups

. Let G be a finite group and ( ) p n G be the number of Sylow p - subgroup of . G In this work it is proved if G is a finite centerless group and

, where r is a prime but not Mersenne prime and 2 r does not divide order of G , then , onde r é um primo mas não um primo de Mersenne e 2 r não divide a ordem de G , então 2 2

Introduction
If n is an integer, then we denote by ( ) n  the set of all prime divisors of .n Let G be a finite group.

Denote by ( ) G
 the set of primes p such that G contains an element of order p .Throughout this paper, we denote by ( ) . All other notations are standard and we refer to (CONWAY et al., 1985), for example.
In 1992, Bi (BI, 1992) showed that 2 ( ) k L p can be characterized only by the order of normalizer of its Sylow subgroups.In other words, if G is a group and . This type of characterization is known for the following simple groups: 2 ( ) k L p (BI, 1992), ( ) n L q (BI, 1995), 4 ( ) S q (BI, 2001), the alternating simple groups (BI, 2001), ( ) , 1990, 2004) for every prime p , then whether G is isomorphic to S ?By the following counterexample it is not always true.
Consider the simple group 5 .
where H is a finite nilpotent group with for every prime p , but G is not isomorphic to 5 .A In Asboei (2013) and Asboei et al. (2011) it is . Also in where r is prime but not Mersenne prime and 2 r does not divide order of .

Preliminary results
In this section we bring two preliminary lemmas used in the proof of the main theorem.
First let / G K isomorphic to 2 ( ) L r .By Lemma 1, On the other hand, if .Then C contains a full Sylow p - subgroup of G for all primes p different from q , and thus | : | G C is a power of q .Now let S be a Sylow q −subgroup of G Then Since by the assumption ( ) 1 Z G  , it follows that 1 Q  .Since q is arbitrary, . We know that for every , p  n G n L r  for every prime p , where r is prime but not Mersenne prime and 2 r does not divide order of G , then G is isomorphic to 2 ( ) L r or Aut( ( ) PGL ( ) L r r  when 1 r   (mod 8).Comparing this type of characterization with characterization by orders of normalizers of Sylow subgroups, it seems that characterization by the number of Sylow subgroups is much stronger than characterization by orders of normalizers of Sylow subgroups.

G
finite group, simple group, Sylow subgroup.Sobre o reconhecimento do grupo simples L 2 (r) pelo número dos sub-grupos de Sylow RESUMO.Vamos supor que G é um grupo finito e ( ) p n G é o número de Sylow p -sub-grupo de.Nessa pesquisa prova-se que, se G é um grupo finite sem centro e

2
Lemma 1 -(ZHANG, 1995)  Let G be a finite group and M be a normal subgroup of G .Then both G M n G for every prime .p Lemma 2 -(BRAUER; REYNOLD, 1958) (Brauer-Reynolds) Suppose G G   and that | | Gis divisible by the prime r but not by 2 .normalizer of a Sylow r - subgroup is the set of upper triangular matrices with determinant 1, so the order of the normalizer is ( 1) r r  .The order of the whole group r -subgroups Asboei et al. (2013)it is proved if G is a finite . Let H be minimal among normal subgroups of G with order divisible by .
 .Let R be a Sylow r -subgroup of , H so R acts coprimely on N .Let p be any prime divisor of | | N , and choose an R -invariant Sylow p -subgroup P of N .If R does not centralizes , P  and r is Mersenne, which is a contradiction.It follows that R centralizes a Sylow p -subgroup of N for every prime p , and thus R centralizes N and hence R is normal in H .This is a contradiction since H has 1 r  Sylow r -subgroups, and the proof is complete.
p p