The Maximal Subgroups of The Symplectic Group Psp(8, 2)

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1 ARPN Journal of Sysems and Sofware AJSS Journal. All righs reserved hp:// The Maximal Subgroups of The Symplecic Group Psp(8, 2) Rauhi I. Elkhaib Dep. of Mahemaics, Faculy of Applied Science, Thamar Universiy, Yemen ABSTRACT The purpose of his paper is o sudy maximal subgroups of he symplecic group PSp(8, 2). The main resul is a lis of maximal subgroups called "he main heorem" which has been proved by using Aschbacher s Theorem ([1]). Thus, his work is divided ino wo main pars: Par (1): In his par, we will find he maximal subgroups in he classes C 1 C 8 of Aschbacher s Theorem ([1]). Par (2): In his par, we will find he maximal subgroups in he class C 9 of Aschbacher s Theorem ([1]), which are he maximal primiive subgroups H of G ha have he propery ha he minimal normal subgroup M of H is no abelian group and simple, hus, we divided his par ino wo cases: Case (1): M is generaed by ransvecions: In his case, we will use resul of Kanor ([2]). Case (2): M is a finie primiive subgroup of rank hree: In his case, we will use he classificaion of Kanor and Liebler ([8]). Mahemaics Subjec Classificaion: 20B05; 20G40, 20H30, 20E28. Keywords: Finie groups; linear groups, marix groups, maximal subgroups. 1. INTRODUCTION The general linear group GL(n, q), consising of he se of all inverible n n marices, so in he marix form, he symplecic group Sp(2n, q) = {g GL(2n, q): g P g = P, where g is he ranspose marix of he marix g and 0 I }. Since, he deerminan of any skew-symmeric marix {A = A} of odd size is zero, hus in he symplecic case, he X1 X2 dimension mus be even. If g =, hen g X3 X4 Sp(2n, q) if and only if X X X X = 0 n n P = or P = diag,..., -In 0n = X X X X and X X X X = I. Thus, 0n In -I 0 n n, n A 0 In B Q In -Q are 0 inv(a ) 0 I, n Q-I, n Q in Sp(2n, q), where A is an inverible n n marix, B is n n symmeric marix, Q is a diagonal marix of 0 s and 1 s, so ha Q 2 = Q and ( Q - I 3 ) 2 = I 3 - Q {see [4] and [5]}. The projecive symplecic group PSp(2n, q) is he quoien group PSp(2n, q) Sp(2n, q) / (Sp(2n, q) Z), where Z is he group of non-zero scalar marices. The group PSp(2m, q) ( = Sp(2m, q) ) is simple, excep for PSp(2, 2), PSp(2, 3) and PSp(4, 2). An elemen T GL(n, q) is called a ransvecion if T saisfies rank(t I n ) = 1 and (T I n ) 2 = 0. The collineaion of projecive space induced by a ransvecion is called elaion. The axis of he ransvecion is he hyperplane Ker(T I n ); his subspace is fixed elemenwise by T, Dually, he cenre of T is he image of (T - I n ). A spli exension ( a semidirec produc ) A:B is a group G wih a normal subgroup A and a subgroup B such ha G = AB and A B = 1. A non-spli exension A.B is a group G wih a normal subgroup A and G/A B, bu wih no subgroup B saisfying G = AB and A B = 1. A group G = AoB is a cenral produc of is subgroups A and B if G = AB and [A, B], he commuaor of A and B = {1}, in his case A and B are normal subgroups of G and A B Z(G). If A B = {1}, hen AoB = AB. Through his aricle, G will denoe PSp(8, 2), unless oherwise saed. G is a simple group of order = and G acs primiively on he poins of he projecive space PG(7, 2) which is a rank 3 permuaion group on PG(7, 2) {see [6]}. The main heorem of his research is he following heorem: Theorem 1.1. Le G = PSp(8, 2). If H is a maximal subgroup of G, hen H isomorphic o one of he following subgroups: 1. G (p), a group sabilizing a poin. This is isomorphic o a group of form 2 7 :(PGL(1, 2) PSp(6, 2)); 2. G (l), a group sabilizing a line. This are isomorphic o a group of form 2 11 :(PGL(2, 2) PSp(4, 2)); 3. G (2-π), a group sabilizing a plane. This are isomorphic o a group of form 2 12 :(PGL(3, 2) PSp(2, 2)); 126

2 4. G (3-π), a group sabilizing a 3-space. This are isomorphic o a group of form 2 10 :PGL(4, 2); ARPN Journal of Sysems and Sofware AJSS Journal. All righs reserved hp:// embedded in GL(n, q o ).Z, where Z is he cenre group of H. 5. PSp(2, 2) PSp(6, 2); 6. H 1 = PSp(2, 2):S 4 ; 7. H 2 = PSp(4, 2):S 2 ; 8. H 3 = PSp(4, 2 2 ).2; 9. H 4 = PSp(2, 2)o PSp(4, 2); 10. H 5 = PSp(2, 2):S 3 ; 11. PSGO + (8, 2); 12. PSGO - (8, 2); 13. PSp(6, 2). We will prove his heorem by Aschbacher s heorem (Resul 2.8) [1]: 2. ASCHBACHER S THEOREM A classificaion of he maximal subgroups of GL(n, q) by Aschbacher s heorem [1], is a very srong ool in he finie groups for finding he maximal subgroups of finie linear groups. There are many good works in finie groups which simplify his heorem, see for example [7]. Bu before saring a brief descripion of his heorem, we will give he following definiions: Definiion 2.1: Le V be a vecor space of dimensional n over a finie field q, a subgroup H of GL(n, q) is called reducible if i sabilizes a proper nonrivial subspace of V. If H is no reducible, hen i is called irreducible. If H is irreducible for all field exension F of F q, hen H is absoluely irreducible. An irreducible subgroup H of GL(n, q) is called imprimiive if here are subspaces V 1, V 2,, V k, k 2, of V such ha V = V 1 V k and H permues he elemens of he se { V 1, V 2,, V k } among hemselves. When H is no imprimiive hen i is called primiive. Definiion 2.2: A group H GL(n, q) is a superfield group of degree s if for some s divides n wih s>1, he group H may be embedded in GL(n/s, q s ). Definiion 2.3: If he group H GL(n, q) preserves a decomposiion V = V 1 V 2 wih dim(v 1 ) dim(v 2 ), hen H is a ensor produc group. Definiion 2.4: Suppose ha n = r m and m > 1. If he group H GL(n, q) preserves a decomposiion V=V 1 V m wih dim(v i ) = r for 1 i m, hen H is a ensor induced group. Definiion 2.5: A group H GL(n, q) is a subfield group if here exiss a subfield F F such ha H can be qo q Definiion 2.6: A p-group H is called a special group if Z(H) = H and is called an exraspecial group if also Z(H) = p. Definiion 2.7: Le Z denoe he cenre group of H. Then H is almos simple modulo scalars if here is a non-abelian simple group T such ha T H/Z Au(T), he auomorphism group of T. A classificaion of he maximal subgroups of GL(n, q) by Aschbacher s heorem [1], can be summarized as follows: Resul 2.8. ( Aschbacher s heorem ) Le H be a subgroup of GL(n, q), q = p e wih he cenre Z and le V be he underlying n-dimensional vecor space over a field q. If H is a maximal subgroup of GL(n, q), hen one of he following holds: C 1 :- H is a reducible group. C 2 :- H is an imprimiive group. C 3 :- H is a superfield group. C 4 :- H is a ensor produc group. C 5 :- H is a subfield group. C 6 :- H normalizes an irreducible exraspecial or symplecic-ype group. C 7 :- H is a ensor induced group. C 8 :- H normalizes a classical group in is naural represenaion. C 9 :- H is absoluely irreducible and H/(H Z) is almos simple. 3. Classes C 1 C 8 of Resul 2.8: In his secion, we will find he maximal subgroups in he classes C 1 C 8 of Resul 2.8: Lemma 3.1: There are hree reducible maximal subgroups of C 1 in G which are: 1. G (p), a group sabilizing a poin. This is isomorphic o a group of form 2 7 :(PGL(1, 2) PSp(6, 2)). 2. G (l), a group sabilizing a line. This are isomorphic o a group of form 2 11 :(PGL(2, 2) PSp(4, 2)). 3. G (2-π), a group sabilizing a plane. This are isomorphic o a group of form 2 12 :(PGL(3, 127

3 2) PSp(2, 2)). VOL. 2, NO. 3, March 2012 ISSN G (3-π), a group sabilizing a 3-space. This are isomorphic o a group of form 2 10 :PGL(4, 2). 5. PSp(2, 2) PSp(6, 2). Proof: Le H be a reducible subgroup of he symplecic group Sp(2n, q) and W be an invarian subspace of H. Le r = dim (W), 1 r n/2 and le G r = G (W) denoe he subgroup of Sp(2n, q) conaining all elemens fixing W as a whole and H G (W). wih a suiable choice of a basis, G (W) consiss of all marices of he form where n = r + m, C is elemenary abelian groups of order q 2rm, A is a p- group of upper riangular marix of order, D GL(r, q), B Sp(2m, q) such ha A P A = P wih 0. Thus he maximal parabolic subgroups are he sabilizers of oally isoopic subspaces <e 1, e 2,, e r > is isomorphic o a group of he form q r(r+1) + 2rm 2 :(GL(r, q) Sp(2m, q)). Also, H is a maximal reducible subgroup of he uniary group Sp(2n, q) which sabilizers of non-singular subspaces of dimension d have he shape H = Sp(2d, q) Sp(2b, q) where n = d + b and d < b. Thus, for PSp(8, 2), here are hree reducible maximal subgroups of G: 1. If r = 1 and m = 3, hen we ge a group G (p), sabilizing a poin. This is isomorphic o a group of form 2 7 :(PGL(1, 2) PSp(6, 2)). 2. If r = 2 and m = 2, hen we ge a group G (l), sabilizing a line. This are isomorphic o a group of form 2 11 :(PGL(2, 2) PSp(4, 2)). ARPN Journal of Sysems and Sofware AJSS Journal. All righs reserved hp:// hus H are isomorphic o Sp(2m, q):s wih 0 < m < n = m, 2. Consequenly, here are wo imprimiive groups of C 2 in PSp(8, 2) which are: 1. If = 4 and m = 1, hen we have H 1 = PSp(2, 2):S 4 ; 2. If =2 and m = 4, hen we have H 2 = PSp(4, 2):S 2. Which proves he poins (6) and (7) of he main heorem 1.1. Lemma 3.3: There is one semilinear group of C 3 in G which is H 3 = PSp(4, 2 2 ).2. Proof: Le H is (superfield group) a semilinear groups of PSp(2n, q) over exension field F r of GF(q) of prime degree r > 1 where r prime number divide n. Thus V is an F r -vecor space in a naural way, so here is an F-vecor space isomorphism beween 2n-dimensional vecor space over F and he m-dimensional vecor space over F r, where m = n/r, hus H embeds in PSp(2m, q r ).r. Consequenly, here is one C 3 group in PSp(8, 2) which is H 3 = PSp(4, 2 2 ).2. This proves he poin (8) of he main heorem 1.1. Lemma 3.4: There is one ensor produc group of C 4 in G which is H 4 = PSp(2, 2)o PSp(4, 2). Proof: If H is a ensor produc group of Sp(2n, q), hen H preserves a decomposiion of V as a ensor produc V 1 V 2, where dim(v 1 ) dim(v 2 ) of spaces of dimensions 2k and 2m over GF(q) and 2n = 4km, k m. So, H sabilize he ensor produc decomposiion F 2k F 2m. Thus, H is a subgroup of he cenral produc of Sp(2k, q)osp(2m, q). Consequenly, here exis one ensor produc group of C 4 in G which is H 4 = PSp(2, 2)o PSp(4, 2). This proves he poin (9) of he main heorem 1.1. Lemma 3.5: There is no subfield group of C 5 in G. 3. If r = 3 and m = 1, hen we ge a group G (2-π), sabilizing a plane. This are isomorphic o a group of form 2 12 :(PGL(3, 2) PSp(2, 2)). 4. If r = 4 and m = 0, hen we ge a group G (3-π), sabilizing a 3-space. This are isomorphic o a group of form 2 10 :PGL(4, 2). 5. If d = 1 and b = 3, hen we ge a group PSp(2, 2) PSp(6, 2). Which proves he poins (1), (2), (3), (4) and (5) of he main heorem 1.1. Lemma 3.2: There are wo imprimiive groups of C 2 in G which are H 1 = PSp(2, 2):S 4 and H 2 = PSp(4, 2):S 2. Proof: If H is imprimiive of he symplecic group Sp(2n, q), hen H preserves a decomposiion of V as a direc sum V = V 1 V, 2, ino subspaces of V, each of dimension m = n/, which are permued ransiively by H, Proof: If H is a subfield group of he symplecic group Sp(2n, q) and q = p k, hen H is he symplecic group over subfield of GF(q) of prime index. Thus H can be embedded in Sp(2n, p f ), where f is prime number divides k. Consequenly, since 2 is a prime number, hen here is no subfield group of C 5 in G. Lemma 3.6: There are no C 6 groups in G. Proof: For he dimension 2n = r m and r is prime of he symplecic group Sp(2n, q). If r is odd prime and r divides q-1, hen H = 2 2m+1. Ω - (2m, 2) normalizes an exraspecial r-group which fixes he symplecic form. Oherwise if r = 2 and 4 divides q - 1, hen H = 2 2m.O - (2m, 2) normalizes an exraspecial 2-group which fixes he symplecic form {see [8]}. Consequenly, here are no C 6 groups in PSp(8, 2) since q = 2 and 4 no divide q - 1. Lemma 3.7: There is one ensor induced group of C 7 in G which is H 5 = PSp(2, 2):S

4 Proof: If H is a ensor induced of he symplecic group Sp(2n, q), hen H preserves a decomposiion of V as V 1 V 2 V r, where V i are isomorphic, each V i has dimension 2m, dim V = 2n = (2m) r, and he se of V i is permued by H, so H sabilize he ensor produc decomposiion F 2m F 2m F 2m, where F = F q. Thus, H/Z PSp(2m, q):s r. Consequenly, here is one ensor induced group of C 7 in G which is H 5 = PSp(2, 2):S 3. This proves he poin (10) of he main heorem 1.1. Lemma 3.8: There are wo maximal C 8 groups in G which are PSGO + (8, 2) and PSGO - (8, 2). Proof: The groups in his class are sabilizers of forms, his means H is he normalizers of one classical groups PSL(2n, q), PO (2n, q) or PSU(2n, q) as a subgroup of PSp(2n, q). Bu from [9] and [10], if q is even, hen he normalizers of PO + (2n, q) and PO - (2n, q) are maximal subgroups of PSp(2n, q) excep when n = 2 and = -. Consequenly, In C 8, here are wo irreducible maximal subgroups in PSp(8, 2) ha are PSGO + (8, 2) and PSGO - (8, 2). Which prove he poins (11) and (12) of heorem 1.1. In he following, we will find he maximal subgroups of class C 9 of Resul 2.8: 4. THE MAXIMAL SUBGROUPS OF C 9 In Corollary 4.1, we will find he primiive non abelian simple subgroups of G. In Theorem 4.2, we will find he maximal primiive subgroups H of G which have he propery ha he minimal normal subgroup M of H is no abelian group and simple. We will prove his Theorem 4.2 by finding he normalizers of he groups of Corollary 4.1 and deermine which of hem are maximal. Corollary 4.1: If M is a non abelian simple group of a primiive subgroup H of G, hen M is isomorphic o one of he following groups: (i) PSO - (8, 2); (ii) PSO + (8, 2); (iii) PΩ - (6, 2); (iv) PΩ + (8, 2); (v) PSp(6, 2); Proof: Le H be a primiive subgroup of G wih a minimal normal subgroup M of H which is no abelian and simple. So, we will discuss he possibiliies of M of H according o: ARPN Journal of Sysems and Sofware AJSS Journal. All righs reserved hp:// To find he primiive subgroups H of G which have he propery ha a minimal normal subgroup of H is no abelian is generaed by ransvecions, we will use he following resul of Kanor {see [2]}: Resul 4.1.1: Le H be a proper irreducible subgroup of Sp(2n, q i ) generaed by ransvecions. Then H is one of: 1. Sp(2n, q); 2. O ± (n, q i ) for q even; 3. S 2n or S 2n+1; 4. SL(2, 5) < Sp(2, 9 i ); 5. Dihedral subgroups of Sp(2, 2 i ). In he following, we will discuss he differen possibiliies of Resul 4.1.1: Corollary 4.1.2: If M is a primiive subgroup of PSp(8, 2) generaed by ransvecions which is no abelian and simple, hen M isomorphic o orhogonal groups PSO - (8, 2) and PSO + (8, 2). Proof: From Resul 4.1.1, M is isomorphic o one of he following groups: 1. From Lemma 3.8, PSO - (8, 2) and PSO + (8, 2) are maximal subgroups of PSp(8, 2). 2. The irreducible 2-modular characers for S 8 by GAP are: [ 1, 1], [ 6, 1], [ 8, 1], [ 14, 1 ], [ 40, 1 ], [ 64, 1 ] (gap> CharacerDegrees(CharacerTable("S8")mod2); ) Thus here is one irreducible characer of degree 8, hus S 8 G, bu he symmeric group S 8 is no a simple. 3. The irreducible 2-modular characers for S 9 by GAP are: [ [ 1, 1 ], [ 8, 1 ], [ 16, 1 ], [ 26, 1 ], [ 40, 1 ], [ 48, 1 ], [ 78, 1 ], [ 160, 1 ] ] (gap> CharacerDegrees(CharacerTable("S9")mod 2);) Thus here is one irreducible characer of degree (I) M conains ransvecions, {secion 4.1}. (II) M is a finie primiive subgroup of rank hree, {secion 4.2}. 4.1 Primiive subgroups H of G which have he propery ha a minimal normal subgroup of H is no abelian is generaed by ransvecions 8, hus S 9 G, bu he symmeric group S 8 is no a simple group. 4. SL(2, 5) G, since he irreducible 2-modular characers for SL(2, 5) by GAP are:[ [ 1, 1 ], [ 2, 2 ], [ 4, 1 ] ] 129

5 gap> VOL. 2, NO. 3, March 2012 ISSN CharacerDegrees(CharacerTable("L2(5)")mod 2); And non of hese characers of degree If M is a Dihedral subgroups of Sp(2, 2 i ), hen M G, since M is no a simple group. 4.2 Primiive subgroups H of G which have he propery ha a minimal normal subgroup M of H which is no abelian is a finie primiive subgroup of rank hree A group G has rank 3 in is permuaion represenaion on he coses of a subgroup K if here are exacly 3 (K, K )-double coses and he rank of a ransiive permuaion group is he number of orbis of he sabilizer of a poin, hus if we consider PSp(2m, q), m 2 and q is of a prime power, as group of permuaions of he absolue poins of he corresponding projecive space, hen PSp(2m, q) is a ransiive group of rank 3. Indeed he poinwise sabilizer of PSp(2m, q) has 3 orbis of lenghs 1, q(q 2m-2 - l)/(q - 1) and q 2m-1 {see [11]}. So we will consider he minimal normal subgroup M of H is no abelian and a finie primiive subgroup of rank hree, so will use he classificaion of Kanor and Liebler {Resul 4.2.2} for he primiive groups of rank hree {see [3]}. secion: The following Corollary is he main resul of his Corollary 4.2.1: If M is a non abelian simple group which is a finie primiive subgroup of rank hree group of H, hen M is isomorphic o one of he following groups: 1. PΩ - (6, 2); 2. PΩ + (8, 2); ARPN Journal of Sysems and Sofware AJSS Journal. All righs reserved hp:// (v) SU(3, 5); (vi) SU(4, 3); (vii) Sp(6, 2); (viii) Ω(7, 3); (ix) SU(6, 2); In he following, we will discuss he differen possibiliies of Resul 4.2.2; Lemma 4.2.3: If M = PSp(4, q), hen M G. Proof: In our case q = 2, bu PSp(4, 2) is no simple. Lemma 4.2.4: If M = PSU(4, q), hen M G. Proof: In our case q = 2, bu PSU(4, 2) G since he irreducible 2-modular characers for PSU(4, 2) by GAP: [ [ 1, 1 ], [ 4, 2 ], [ 6, 1 ], [ 14, 1 ], [ 20, 2 ], [ 64, 1 ] ]. (gap>characerdegrees(characertable("u4(2)")mod 2); ) and non of hese characers of degree 8. Lemma 4.2.5: PSU(5, q) G. Proof: PSU(n, q), n 3, has no projecive represenaion in G of degree less han q(q n-1-1)/(q+1), if n is odd {see [12] and [13]}, hus PSU(5, q), has no projecive represenaion in G for all q 3, hus PSU(5, q) G. Lemma 4.2.6: PΩ - (6, 2) G. Proof: In our case q = 2, hus we will consider PΩ - (6, 2)? PSp(6, 2) G since he irreducible 2-modular characers for PSp(6, 2) by GAP are: [ [ 1, 1 ], [ 6, 1 ], [ 8, 1 ], [ 14, 1 ], [ 48, 1 ], [ 64, 1 ], [ 112, 1 ], [ 512, 1 ] ] (gap>characerdegrees(characertable("s6(2)")mod 2); ) 3. PSp(6, 2); Proof:Le M is no an abelian and is a finie primiive subgroup of rank hree of H, and will use he classificaion of Kanor and Liebler {Resul 4.2.2} for he primiive groups of rank hree {see [3]}. So, we will prove Corollary by series of Lemmas hrough Lemmas and Resul Resul 4.2.2: If Y acs as a primiive rank 3 permuaion group on he se X of coses of a subgroup K of Sp(2n-2, q), Ω ± (2n, q), Ω(2n-1, q) or SU(n, q). Then for n 3, Y has a simple normal subgroup M *, and M * Y Au(M * ), where M * as follows: (i) M = Sp(4, q), SU(4, q), SU(5, q), Ω - (6, q), Ω + (8, q) or Ω + (10, q). (ii) M = SU(n, 2), Ω ± (2n, 2), Ω ± (2n, 3) or Ω(2n-1, 3). (iii) M = Ω(2n-1, 4) or Ω(2n-1, 8); (iv) M = SU(3, 3); Bu PΩ - (6, 2) PSp(6, 2), hen PΩ - (6, 2) G. Lemma 4.2.7: PΩ + (8, 2) G. Proof: See Lemma 3.8. Lemma 4.2.8: PΩ + (10, q) G. Proof: PΩ + (2n, q), n 4, q = 2, 3 or 5, has no projecive represenaion in G of degree less han q n-2 ( q n-1-1), {see [12] and [13]}, hus, q = 2, and n = 5, hen his bound is greaer han 8, hus PΩ + (10, 2) G. Lemma 4.2.9: if M = PSU(n, 2), hen M G. Proof: In our case, 2n-2 = 8, hus n = 5 and we need o consider PSU(5, 2)? The irreducible 2-modular characers for PSU(5, 2) by GAP are: [ 1, 1 ], [ 5, 2 ], [ 10, 2 ], [ 24, 1 ], [ 40, 4 ], [ 74, 1 ], [ 160, 2 ], [ 280, 2 ], [ 1024, 1 ] 130

6 (gap>characerdegrees(characertable("u5(2)")mod 2);) ARPN Journal of Sysems and Sofware AJSS Journal. All righs reserved hp:// ( gap > G:=PSp(8, 8);; And none of hese characers of degree 8.Thus PSU(5, 2) G Lemma : If M = PΩ ± (2n, 2), hen M G. Proof: In our case n = 5, hus we need o consider PΩ ± (10, 2): PΩ + (2n, q), n 4, q = 2 has no projecive represenaion in G of degree less han q n-2 ( q n-1-1), {see [12] and [13]}, bu his bound is greaer han 8, hus PΩ + (10, 2) G. PΩ - (2n, q), n 4, has no projecive represenaion in G of degree less han (q n-1 + 1)( q n-2-1), {see [12] and [13]}, bu his bound is greaer han 8, hus PΩ - (10, 2) G. Lemma : If M = PΩ ± (2n, 3), hen M G. Proof: In our case n = 5, hus we need o consider PΩ ± (10, 3): PΩ + (2n, q), n 4, q = 2 has no projecive represenaion in G of degree less han q n-2 ( q n-1-1), {see [12] and [13]}, bu his bound is greaer han 8, hus PΩ + (10, 3) G. PΩ - (2n, q), n 4, has no projecive represenaion in G of degree less han (q n-1 + 1)( q n-2-1), {see [12] and [13]}, bu his bound is greaer han 8, hus PΩ - (10, 3) G. Lemma : If M = PΩ(2n-1, 3), hen M G. Proof: In our case n = 5, hus, we have PΩ(9, 3) G, since PΩ(2n+1, q), n 3, q = 3, has no projecive represenaion in G of degree less han q n-1 (q n-1-1), {see [12] and [13]}, which is greaer han 8. Lemma : If M = PΩ(2n-1, 4), hen M G. Proof: In our case n = 5, hus we have PΩ(9, 4) PSp(8, 4), bu he order of PSp(8,4) is equal o (gap > G:= PSp(8, 4) ;; gap> Order(G); ) This is no dividing he order of G. Thus PΩ(9, 4) G Lemma : If M = PΩ(2n-1, 8), hen M G. Proof: In our case n = 5, hus we have PΩ(9, 8) PSp(8, 8), bu he order of PSp(8, 8) is equal o gap> Order(G); ) which is no dividing he order of G. Thus PΩ(9, 8) G. Lemma : PSU(3, 3) G. Proof: The irreducible 2-modular characers for PSU(3, 3) by GAP are: [ [ 1, 1 ], [ 6, 1 ], [ 14, 1 ], [ 32, 2 ] ], gap>characerdegrees(characertable("u3(3)")mod 2); And none of hese of degree 8. Thus PSU(3, 3) G. Lemma : PSU(3, 5) G. Proof: Since he irreducible 2-modular characers for PSU(3, 5) by GAP are: [ [ 1, 1 ], [ 20, 1 ], [ 28, 3 ], [ 104, 1 ], [ 144, 2 ] ] (gap>characerdegrees(characertable("u3(5)")mod 2); ) And none of hese characers of degree 8. Lemma : PSU(4, 3) G. Proof: Since he irreducible 2-modular characers for PSU(4, 3) by GAP are: [ [ 1, 1 ], [ 20, 1 ], [ 34, 2 ], [ 70, 4 ], [ 120, 1 ], [ 640, 2 ],[ 896, 1 ] ] (gap>characerdegrees(characertable("u4(3)")mod 2); ) And non of hese characers of degree 8. Lemma : PSp(6, 2) G. Proof: See Lemma Lemma : PΩ(7, 3) G. Proof: Since PΩ(2n+1, q), n 3, has no projecive represenaion in G of degree less han q n-1 (q n-1-1), {see [12] and [13]}, hus for q = 3 and n = 3, PΩ(7, 3) has a projecive represenaion of degree which is greaer han 8. Lemma : PSU(6, 2) G. Proof: Since he irreducible 2-modular characers for PSU(6, 2) by GAP are: [ [ 1, 1 ], [ 20, 1 ], [ 34, 1 ], [ 70, 2 ], [ 154, 1 ], [ 400, 1 ], [ 896, 2 ], [ 1960, 1 ], [ 3114, 1 ], [ 32768, 1 ] ] (gap>characerdegrees(characertable("u6(2)")mod 2); ) And non of hese of degree 8. Now, we will deermine he maximal primiive group of C 9 : Theorem 4.2: If H is a maximal primiive subgroup of G which has he propery ha a minimal normal subgroup M of H is no abelian group, hen H is isomorphic o one of he following subgroups of G: 131

7 (i) PSGO - (8, 2); (ii) PSGO + (8, 2); (iii) PSp(6, 2); VOL. 2, NO. 3, March 2012 ISSN Proof: We will prove his heorem by finding he normalizers N of he groups of Corollary 4.1 and deermine which of hem are maximal: From [14], he normalizer of Sp(2n, k ) in SL(2n, k) is SGSp(2n, k) = GSp(2n, k) SL(2n, k). From [10], he normalizer of SO(n, k ) in SL(n, k) is SGO(n, k) = GO(n, k) SL(n, k). Thus, If Y = PSO - (8, 2), hen N = PSGO - (8, 2), which prove he poin (11) of heorem 1.1. If Y = PSO + (8, 2), hen N = PSGO + (8, 2), which prove he poin (12) of heorem 1.1. If Y = PΩ - (6, 2), hen N = PSGO - (6, 2), which is no a maximal in G since PSGO - (6, 2) is a subgroup of PSGO - (8, 2). If Y = PΩ + (8, 2), hen N = PSGO + (8, 2), which prove he poin (12) of heorem 1.1. If Y = PSp(6, 2), hen N = PSGSp(6, 2) bu in PSp(8, 2), we ge PSGSp(6, 2) = PSp(6, 2), which prove he poin (13) of heorem 1.1. This complees he proof of heorem 1.1. REFERENCES 1. Aschbacher M., On he maximal subgroups of he finie classical groups, Inven. Mah. 76 (1984), Kanor W. M., Subgroups of classical groups generaed by long roo elemens. Trans. Amer. Mah. Soc. 248 (1979), pp Kanor W. M. and Liebler R. A., The Rank 3 Permuaion Represenaions of he Finie Classical Groups, Trans. Amer. Mah. Soc., Vol. 271, No. 1 (1982), Chahal J. S., Arihmeic subgroups of he symplecic group, Osaka J. Mah. Volume 14, Number 3 (1977), ARPN Journal of Sysems and Sofware AJSS Journal. All righs reserved hp:// 5. Hua L. K. and Reiner L., Generaion of he symplecic modular group, Trans. Amer. Mah. Soc. 65 (1949), Cameron P. J., Noes on Classical Groups, (available a: hp:// 7. Kleidman P.B., M. Liebeck, The Subgroup Srucure of he Finie Classical Groups, London Mah. Soc. Lecure Noe Series 129, Cambridge Universiy Press, Wilson R. A, Finie simple groups. ( available a: hp:// ~raw/fsgs.hml ). 9. Dye R. H., Maximal Subgroups Of GL(2n,K), SL(2n,K), PGL(2n,K) And PSL(2n,K) Associaed Wih Symplecic Polariies, J. Algebra 66 (1980), King O. H., On Subgroups of The Special Linear Group Conaining The Special Orhogonal Group, J. Algebra, 96 (1985), Yanushka A., A Characerizaion of The Symplecic groups PSp(2m, q) As Rank permuaion Group, Pacific Journal Of Mahemaics, Vol. 59, No. 2 (1975), Landazuri V. and Seiz G. M., On he minimal degree of projecive represenaions of he finie chevalley groups, J. Algebra 32 (1974), Seiz G. M. and Zalesskii A. E., On he minimal degree of projecive represenaions of he finie chevalley groups, II., J. Algebra 158 (1993), Dye R. H., On he maximaliy of he orhogonal groups in he symplecic groups in characerisic wo, Mah. Z. 172 (1980), GAP program. version 4.4. ( available a: hp:// ). 132