NORMALIZERS AS THE MATHIEU GROUPS

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NORMALIZERS AS THE MATHIEU GROUPS

BEHROOZ KHOSRAVI AND BEHNAM KHOSRAVI Received 17 October 2004

There exist many characterizations for the sporadic simple groups. In this paper we give two new characterizations for the Mathieu sporadic groups. LetMbe a Mathieu group and letpbe the greatest prime divisor of|M|. In this paper, we prove thatMis uniquely determined by|M|and|NM(P)|, whereP∈Syl_p(M). Also we prove that ifGis a finite group, thenG^∼=M if and only if for every prime q,|NM(Q)| = |NG(Q)|, whereQ∈ Syl_q(M) andQ∈Syl_q(G).

1. Introduction

LetG be a finite group. We denote byπ(G) the set of all prime divisors of |G|. It was proved that ifGis an alternating group, a finite projective special linear group, a Janko sporadic simple group, or a finite projective special symplectic group, thenGis characterizable by the orders of normalizers of its Sylow subgroups [1,2,3,4,10].

Mazurov and Shi [11,12,13,14] and Deng [7] proved that some of the almost sporadic simple groups are characterizable by the set of element orders. Chen [5] and A. Khosravi [9] proved that some of the almost sporadic simple groups are characterizable by the set of order components.

Notation 1. For a prime numberq, we define

nq(G)=NG(Q) whereQ∈Syl_q(G). (1.1) In this paper, as the main result, the following theorems are proved.

Theorem1.1. LetM be a Mathieu group and letG be a finite group. If for every prime numberq,nq(M)=nq(G), thenG^∼=M.

Theorem1.2. LetMbe a Mathieu group and letGbe a finite group. Letpbe the greatest prime divisor of|M|. If|G| = |M|andnp(M)=np(G), thenG^∼=M.

In this paper, all groups are finite. All further unexplained notations are standard and refer to [6], for example. Letmbe a positive integer and letqbe a prime number. Then mqdenotes theq-part ofm. In other words,mq=q^kifq^km(i.e.,q^k|mbutq^k+1m).

International Journal of Mathematics and Mathematical Sciences 2005:9 (2005) 1449–1453 DOI:10.1155/IJMMS.2005.1449

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We would like to note that|Z6| = |S3| =6,Z3Z6, andZ3S3. Therefore|NZ6(P)| =

|NS3(P)| =6, whereP^∼=Z3is the 3-Sylow subgroup ofZ6andS3. ButZ6∼=S3and hence they are not characterizable by these conditions.

2. Preliminary results

The following lemma is an immediate consequence of [15, Theorem 2.1.17].

Lemma2.1. LetGbe aq-group and let|G| =q^k, for somek >0. Thenpdivides^k_i₌1(qⁱ− 1), for any prime divisorpof|Aut(G)|withp=q.

Lemma2.2 [8]. The following results hold:

(i)|M11| =7920,n2(M11)=16,n3(M11)=144,n5(M11)=20, andn11(M11)=55;

(ii)|M12| =95 040,n2(M12)=64,n3(M12)=108,n5(M12)=40, andn11(M12)=55;

(iii)|M22| =443 520,n2(M22)=128,n3(M22)=72,n5(M22)=20,n7(M22)=21, and n11(M22)=55;

(iv)|M23| =10 200 960,n2(M23)=128,n3(M23)=144,n5(M23)=60,n7(M23)=42, n11(M23)=55, andn23(M23)=253;

(v)|M24| =244 823 040,n2(M24)=1024,n3(M24)=216,n5(M24)=240,n7(M24)= 126,n11(M24)=110, andn23(M24)=253.

Remark 2.3. Note that we can calculate the crucial normalizer ordersn11(M11),n11(M12), n11(M22),n23(M23), andn23(M24) without using GAP. For example, from the character table in atlas [6], we can see that there are two conjugacy classes of elements of order 11 inM11, each of order 720, for a total of 1440 elements of order 11. Since a Sylow 11- subgroup ofM11has order 11, that means there are 144 Sylow 11-subgroups. Therefore, n11(M11)= |M11|/144=55. Similar calculations work in the other cases.

Lemma2.4. LetMbe a Mathieu group and letGbe a finite group. If for every prime number q,nq(M)=nq(G), then|G| = |M|.

Proof. Obviously ifQis aq-Sylow subgroup ofG, thenQ≤NG(Q) and hence|Q|divides

|NG(Q)|. Also|G|q= |Q|. Sincenq(G)=nq(M), we conclude that (nq(G))q=(nq(M))q. Therefore for every primeq, we have|G|q= |M|q. Hence|G| = |M|. Remark 2.5. Lemma 2.4 shows that ifTheorem 1.2 is proved, thenTheorem 1.1 is an immediate consequence of Theorem 1.2. Therefore in the next section we only prove Theorem 1.2.

3. Characterizations of the Mathieu groups

We note that in the proof ofTheorem 1.2, we use the classification of finite simple groups.

We prove this theorem by using the following lemmas.

Lemma 3.1. Let Gbe a finite group and let M be M11,M12, or M22. If |G| = |M|and n11(M)=n11(G), thenG has a normal series1HKGsuch thatK/H is a simple group,11is a divisor of|K/H|, and|G/K| |5.

Proof. Let 1=G0≤G1≤G2≤ ··· ≤Gk=Gbe a chief series ofG. Since 11 |M|, there exists someisuch that 11 is a divisor of|Gi+1/Gi|. So letGi=HandGi+1=K. SinceK/H

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is a chief factor ofG, it follows thatK/H is a minimal normal subgroup ofG/H. Also K/H is characteristically simple which implies thatK/Hbe a simple group or a product of isomorphic simple subgroups. We know that 11| |K/H|, but 11²|G/H|and hence K/His a simple group.

Therefore 1HKGis a normal series ofG, such thatK/His a simple group and 11 is a divisor of|K/H|. LetP be a 11-Sylow subgroup ofG. ThereforeP≤K, which implies thatG=KNG(P), by the Frattini argument. Then

G

K ^∼⁼KNG(P)

K ^∼⁼ NG(P)

NG(P)∩K ^∼⁼NG(P)

NK(P). (3.1)

Therefore|G/K|is a divisor of|NG(P)| =55. Since 11 divides|K/H|, we conclude that

|G/K|divides 5 and the proof is complete.

Lemma 3.2. Let Gbe a finite group and let M be M11,M12, or M22. If |G| = |M|and n11(M)=n11(G), thenG^∼=M.

Proof. ByLemma 3.1,Ghas a normal series 1HKGsuch thatK/H is a simple group, 11 is a divisor of|K/H|, and|G/K|divides 5. We claim that|H|is a divisor of 5.

Since 11 divides|K/H|, we have 11|H|. Ifq∈π(H), then letQbe aq-Sylow subgroup of H. NowHKandQ∈Syl_q(H). Hence by using the Frattini argument, we conclude that K=HNK(Q). We know thatCK(Q)NK(Q) andNH(Q)NK(Q). ThereforeNH(Q) CK(Q)NH(Q)NK(Q). Also

K

H ^∼⁼HNK(Q)

H ^∼⁼ NK(Q) H∩NK(Q)^∼⁼

NK(Q)

NH(Q), (3.2)

andK/H is a simple group. SoCK(Q)NH(Q)=NH(Q) orCK(Q)NH(Q)=NK(Q). Now we consider these cases separately.

Case 1. IfCK(Q)NH(Q)=NH(Q) and|Q| =q^k, thenCK(Q) is a subgroup of NH(Q).

CK(Q)|. We know thatNK(Q)/CK(Q) is isomorphic to a subgroup of Aut(Q). Also 11|

|K/H|, 11|H|,q∈π(H), and|Q| =q^k (whereQ∈Syl_q(H)) which implies that 11| _k

i=1(qⁱ−1), byLemma 2.1.

But easy calculations show that 11⁷_i₌1(2ⁱ−1), 11³_i₌1(3ⁱ−1), 11(5−1), and 11(7−1). Therefore this case is impossible.

Case 2. IfCK(Q)NH(Q)=NK(Q), then KH ^∼⁼NK(Q)

NH(Q)^∼⁼CK(Q)NH(Q)

NH(Q) ^∼⁼ CK(Q)

CK(Q)NH(Q). (3.3) Since 11 divides |K/H|, we conclude that 11 divides|CK(Q)|, and henceP≤CK(Q), whereP∈Syl₁₁(G). ThereforeQ≤CK(P)≤NG(P). Hence|Q|is a divisor of|NG(P)| = 55. Since 11|Q|, we conclude that|Q|is a divisor of 5, and hence|H| |5.

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Therefore|G/K| · |H|is a divisor of 5. If|G/K| · |H| =5, thenK/H is a simple group of order|M|/5, which is a contradiction by the classification of finite simple groups (see [6, pages 239–241]). (There exists no simple group of order 1584, 19 008, or 88 704.) So|G/K| =1,|H| =1, which implies thatGbe a simple group of orderM. Since there exists only one simple group of order|M|, it follows thatG^∼=M. Lemma3.3. LetGbe a finite group and letMbeM23orM24. If|G| = |M|andn23(M)= n23(G), thenG^∼=M.

Proof. Similar toLemma 3.1we conclude thatGhas a normal series 1HKGsuch thatK/His a simple group, 23 is a divisor of|K/H|, and|G/K|is a divisor of 11. Since 23| |K/H|and 23²|G|, it follows that 23|H|. Also 23¹⁰_i₌1(2ⁱ−1), 23³_i₌1(3ⁱ−1), 23(5−1), 23(7−1), and 23(11−1). Therefore ifQis aq-Sylow subgroup ofH, then

|Q|dividesn23(G)=253. Hence|H|divides 11. Therefore|G/K| · |H|is a divisor of 11. If

|G/K| · |H| =11, thenK/H is a simple group of order|M|/11, which is a contradiction by the classification of finite simple groups (see [6, pages 239–241]). (There exists no simple group of order 927 360 or 222 566 40.) So|G/K| =1,|H| =1, which implies that Gbe a simple group of orderM. Since there exists only one simple group of order|M|, it

follows thatG^∼=M.

Acknowledgments

The authors express their gratitude to Professor Bell and the referee for useful sugges- tions which improved the manuscript. The first author would like to thank the Institute for Studies in Theoretical Physics and Mathematics (IPM) for the financial support. Also we want to thank Professor Thomas Breuer for his help in computing the orders of normalizers of Sylow subgroups of the Mathieu sporadic simple groups. We dedicate this paper to our parents: Professor Amir Khosravi and Mrs. Soraya Khosravi for their un- ending love. The first author was supported in part by a Grant from IPM (no. 82200031).

We also have to thank the authors of the atlas [6] for the rich mine of information they have provided.

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Behrooz Khosravi: Department of Pure Mathematics, Faculty of Mathematics and Computer Sci- ence, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue, Tehran 15914, Iran; Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran

E-mail address:[email protected]

Behnam Khosravi: Department of Mathematics, Faculty of Mathematical Sciences, Shahid Be- heshti University, Evin, Tehran 19838, Iran

E-mail address:[email protected]

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