SOME FINITE SIMPLE GROUPS OF LIE TYPE C n ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER

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1 Joural of Algebra, Nuber Theory: Advaces ad Applicatios Volue, Nuber, 010, Pages SOME FINITE SIMPLE GROUPS OF LIE TYPE C ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER School of Matheatics South-Chia Noral Uiversity Guagzhou, P. R. Chia e-ail: xuigchu00@yahoo.co.c Abstract For a fiite group G, deote by ω ( G) the set of the eleet orders i G. We show the validity of a cojecture of Shi for the fiite siple groups of Lie type C ( q), where q is a eve prie power ad 16. Theore. Let G be a fiite group ad M be oe of the fiite siple groups of Lie type C ( q), where q is a eve prie power ad 16. If G = M ad ω ( G) = ω( M ), the G M. 1. Itroductio For a fiite group G, deote by ω ( G) the set of eleet orders i G. A fiite group G is said to be recogizable by the set of its eleet orders 010 Matheatics Subject Classificatio: 0D05, 0D60. Keywords ad phrases: quasirecogizable group, set of eleet orders, siple group, prie graph. This project is supported by the Natioal Natural Sciece Foudatio of Chia (Grat No ). Received Septeber 1, Scietific Advaces Publishers

2 58 (briefly, recogizable), if every fiite group H with ω ( G) = ω( H ) is isoorphic to G. A fiite o-abelia siple group S is said to be quasirecogizable, if every fiite group H with ω ( S) = ω( H ) has a uique o-abelia copositio factor ad this factor is isoorphic to S. We ca apply the Grueberg-Kegel theore (see Lea.1), whe establishig the certai quasirecogizable property of these groups with discoected prie graphs. I [1, ], it was show that the fiite siple o-abelia groups P with the uber of coected copoets s ( P ) 3, are quasirecogizable except whe P is isoorphic to group A 6. I [10], it was proved that the siple group P, with the uber of coected copoets s ( P ) =, is quasirecogizable whe P is oe of siple groups D ( ), D ( )( > 1), ad C ( )( > ). k + 1 We ca ot apply Lea.1 whe establishig the certai quasirecogizable property of the siple groups C ( q), where q is a eve prie power ad 16, which have coected prie graphs. But, Lea. allows us to start provig quasirecogizability uder uch weaker coditios o the group uder cosideratio. I this way, the siple liear groups L ( ) with = l 16 (see [5, 13]) were proved to be recogizable. We obtai a list of the fiite siple groups S with t ( S) 1 ad t (, S) 3 i this paper. Shi i [8] put forward the followig cojecture: Cojecture. Let G be a fiite group ad M be a fiite siple group. The G M, if ad oly if G = M ad ω ( G) = ω( M ). This cojecture is correct for Z p, where p is a prie uber, A, where 5, sporadic siple groups, siple groups of Lie type except B ( q), C ( q), D q 15 where q is a prie power, ad siple ad ( )[ ], groups with order less tha We shall show the validity of this cojecture for the fiite siple groups of Lie type C ( q), where q is a eve prie power ad 16. k

3 SOME FINITE SIMPLE GROUPS OF LIE TYPE 59 Theore. Let G be a fiite group ad M be oe of the fiite siple groups of Lie type C ( q), where q is a eve prie power ad 16. If G = M ad ω ( G) = ω( M ), the G M. All groups cosidered i this paper are fiite groups ad siple groups are o-abelia. All the uexplaied otatios i this paper are stadard ad ca be foud i [].. Preliiary Results Let G be a group ad ω ( G) be the set of orders of eleets i G. This set is closed ad partially ordered by divisibility, ad hece is uiquely deteried by the set µ ( G) of its eleets, which are axial uder divisibility relatio. If p is a prie, the the p-period of G is the axial power of p that belogs to ω ( G). Let π ( G) be the set of all prie divisors of order of G. The set ω ( G) of the group G defies a prie graph GK ( G), whose vertex set is π ( G) ad two distict pries p, q π( G) are adjacet, if pq ω( G). Deote by s ( G) the uber of coected copoets i GK ( G) ad by π = ( G), i π i i = 1,,, s( G), the i-th coected copoet. If G is eve, the π ( G) will be the coected copoet of G cotaiig. Guided by 1 give graph coceptio, we say that prie divisors p ad r of the order of G are adjacet, if vertices p ad r are joied by edge i GK ( G). Otherwise, pries p ad r are said to be oadjacet. Deote by t ( G) the axial uber of pries i π ( G) pairwise oadjacet i GK ( G). I other words, if ρ ( G) is soe idepedet set with the axial uber of vertices i GK ( G) (the subset of vertices of a graph is called a idepedet set, if its vertices are pairwise oadjacet), the t ( G) = ρ ( G). I graph theory, this uber is called the idepedet uber of a graph. By aalogy, we deote by t (, G) the axial uber of

4 60 vertices i the idepedet sets of GK ( G) cotaiig. If ρ (, G) is soe idepedet set with the axial uber of vertices i GK ( G) cotaiig, the t(, G) = ρ(, G). We call this uber the -idepedet uber. Grueberg ad Kegel gave the followig descriptio for fiite groups with discoected prie graph (see Theore A i [1]): Lea.1 (Grueberg ad Kegel). If G is a fiite group with discoected prie graph, the oe of the followig stateets holds: i (a) s ( G) =, G is Frobeius group; (b) s ( G) =, G = ABC is soluble, where A, AB are oral subgroups G, B is a oral subgroup i BC, ad groups; (c) G = P is o-abelia siple; AB, BC are Frobeius (d) G is a extesio of a π1 ( G)- group by a siple group P; (e) G is a extesio of a group of type (c) or (d) by a π1 ( G)- group. I cases (d) ad (e), GK ( P ) is discoected, s( P ) s( G), ad for each i, there exists j such that π i ( G) = π j ( P ). I this article, we eed the followig stateet, which ca be applied to a wide class of fiite groups icludig the groups with coected Grueberg-Kegel graph (see [1]). Lea.. Let G be a fiite group satisfyig two coditios: (a) there exist three pries i π ( G), which are pairwise oadjacet i GK ( G), that is, t ( G) 3; (b) there exists a odd prie i π ( G), which is oadjacet to prie i GK ( G), that is, t (, G).

5 SOME FINITE SIMPLE GROUPS OF LIE TYPE 61 The, there exists a fiite o-abelia siple group S such that, S G = G K Aut( S) for axial soluble subgroup K of G. Furtherore, t ( S) t( G) 1 ad oe of the followig stateets holds: (1) S Alt7 or L ( q) for soe odd q ad t ( S) = t(, S) = 3; () for every prie p i π ( G) oadjacet to i GK ( G), the Sylow p-subgroup of G is isoorphic to the Sylow p-subgroup of S. I particular, t(, S) t(, G). Proof. See the ai theore i [1]. We use the followig uber-theoretic otatios. If is a atural uber, the π ( ) is the set of prie divisors of. If p π( ), the p is the axial p-power that divides. If q is a atural uber, r is a odd prie, ad ( q, r) = 1, the by e ( r, q), we deote the sallest atural uber such that q 1( odr). Give a odd q, put e (, q) = 1, if q 1( od ) ad put e (, q) =, if q 1( od ). The followig uber-theoretic result is iportat for ivestigatios o the structure of prie graphs of the fiite siple groups of Lie type. Lea.3. Let q, s be atural ubers, q, s. The oe of the followig stateets holds: (1) there exists a prie r, which divides q 1, but does ot divides t q 1 for all atural ubers t < s; () s = 6 ad q = ; (3) s = ad q 1 for soe atural uber. = s Proof. See [16]. The prie r with e ( r, q) = s is called a priitive prie divisor of s s q 1. If q is fixed, we deote by r s ay priitive prie divisor of q 1 s (obviously, q 1 ca have ore tha oe priitive prie divisor).

6 6 We obtai the followig table of the fiite siple groups S with t ( S) 1 ad t (, S) 3. Lea.. Let S be a fiite siple o-abelia group with t ( S) 1 ad t (, S) 3. The S, t(, S), ρ (, S), ad t( S) are as i Table 1. Proof. See the table i [5]. Lea.5. Let G be a fiite group, N G such that G N is a Frobeius group with kerel F ad cyclic copleet C. If ( F, N ) = 1 ad F is ot cotaied i NC G ( N ) N, the p C ω( G) for soe prie divisor of N. Proof. See Lea 1 i [7].

7 SOME FINITE SIMPLE GROUPS OF LIE TYPE 63 Table 1. Siple o-abelia groups S with t ( S) 1 ad t (, S) 3 S Additioal coditios o S t (, S) ρ (, S) \ { } t ( S) Alt, are pries 3 {, } 103 1, 3 are pries 3 { 1, 3} ( q) < ( q 1) = { r 1, r } [ ] A 1 3 q eve 3 { r 1, r } A 1 ( q) < ( q + 1) = { r, r } [ ] 3 0 ( od ) 3 { r, r } 1 ( od ) 3 { r 1, r } ( od ) 3 { r } r, 3 ( od ) 3 { ( ) r } r 1, C ( q), 15 q eve { r, r } [ ] D ( q) q 5 ( od 8), 1 ( od ) 3 { r, r } [ ] 16 0 ( od ) 3 { r, r 1} 1 ( od ) 3 { r, r } D ( q) q 3 ( od 8 ), 1 ( od ) { r, r } [ ] 15 0( od ) 3 { r 1, r, r } 1 ( od ) 3 { r, r }

8 6 Lea.6. The fiite siple group A ( q), > 3, q icludes a Frobeius subgroup with kerel of order order q 1. q ad cyclic copleet of Proof. See Lea.3 i [10]. 3. Proof of the Theore Let L = C ( q), where 16 ad q = k. By Lea., t ( L) 13, t (, L) 3, ad the -period of L is equal to, where is the sallest atural uber such that, i.e., = 1 + log. Let G be fiite group with ω ( G ) = ω( L), G = L, ad K be the axial oral soluble subgroup of G. By Lea., there exists a fiite o-abelia siple group S such that S G = G K Aut( S); oreover, t ( S) t( G) 1 ad either t ( S) = t(, S) = 3 or t (, S) t (, G). Sice t ( G) = t( L) 13, t(, G) = t(, L) 3, ad the -period of L is equal to, where = 1 + log, the group S ust satisfy t ( S) 1, t (, S) 3, ad the -period of S, where = 1 + log. By Lea., S is oe of the groups i Table 1. The proof relies o aalysis of all possibilities for S fro this table case by case. If ot specified, r ad r are soe fixed priitive prie divisors of q 1 ad q 1, respectively. By the defiitio of priitive prie, these ubers are pairwise distict. By Lea., the pries r ad r are oadjacet to i GK ( L), ad so i GK ( G) as well. Cosequetly, by Lea., these pries divide the order of S. (1) S is ot isoorphic to Alt. Let S = Alt. The 103 ad there are two pries aog ubers, 1,, 3; these are r, r. By Lea., we have r ω( L) ad r ω( L). Suppose that r divides the

9 SOME FINITE SIMPLE GROUPS OF LIE TYPE 65 order of S. Sice S does ot cotai a eleet of order r, it follows that r 5. The, there are three pries aog the six cosecutive ubers, 1,, 5; which is ipossible sice 103. So r lies i π( K ) π( Out( S) ). Sice π( Out ( S) ) = { }, we have r π( K ). Deote r by r. Let G = G Or ( K ) ad K = K Or ( K ). The R = O r ( K ) 1. Suppose that K = R. The group S acts faithfully o K. Otherwise, by its siplicity S cetralizes K ad, therefore, G cotais a eleet of order r. The group Alt 6, ad so S as well, icludes a Frobeius group F with kerel of order 9 ad cyclic copleet of order. By applyig Lea.5 to the preiage of F i G, we fid that r ω( G); a cotradictio. Suppose that K R. The, there is a prie t such that T = O ( K R ) is otrivial. Sice O r ( ) = 1, the group T t acts faithfully o R. The T acts faithfully o R = R Φ ( R), where Φ ( R) is the Frattii subgroup of R, as well. Deote by G, the factor group G Φ ( R). By Table 8 i [11], at least oe of the pries r ad r is oadjacet to t i ω ( G). Deote this prie by s. Let x be a eleet of order s i G R. The H = T x is a Frobeius subgroup i G R. The preiage of H i G satisfies coditios of Lea.5, hece G cotais a eleet of order r s, which cotradicts Table 8 i [11], i.e., the pries r ad s are oadjacet i ω ( G). K () S is ot isoorphic to A ε ( 1 q ), where q is odd. Let S A ε = ( 1 q ), where q is odd. The = ( q ε1) > ad t ( S) = ( + 1). Sice t ( S) t( G) 1 ad t ( G) = [ ], we have

10 Let = log. ( + 1) [ ] 1. Hece, [ ] , 1 Sice > S icludes a cyclic subgroup of order q 1. By we have 1 1 q 1 = ( q 1)( q + 1)( q + 1) ( q + 1), ( 1) ( 1) ( 1) ( 1 1 1) ( q = q q + q + q + 1) =. + 1 Thus ω( S). Sice the -period of S, where = 1 + log, 1 + log 1 + log. Therefore, +1, which cotradicts [ ] 3. (3) S is ot isoorphic to D ε ( q ), where q is odd. ε Let S = D ( q ), where q is odd. The is odd ad t ( S) 3 + [ ] Sice t ( S) t( G) 1 ad t ( G) = [ ], we have [ ] [ ] 1. Hece. Suppose that + 3. As S icludes the uiversal coverig of A ( q ), by repeatig the arguets of the preceedig paragraph, we get a cotradictio. Next, suppose that =, 1,, + 1, +. As S icludes the uiversal coverig of A ( q ), by repeatig the arguets of the preceedig paragraph, we get a cotradictio. By we have 1 1 q 1 = ( q 1)( q + 1)( q + 1) ( q + 1), ( 1) ( 1) ( 1) ( 1 1 1) ( + q = q q + q + q + 1) = = Thus ω( S), where = log, ad the -period of S is 1+.

11 SOME FINITE SIMPLE GROUPS OF LIE TYPE 67 We clai K = 1. Suppose that K. Let G = G O ( K ) ad K = K O ( K ). The R = O( K ) 1. The group S acts faithfully o R. Otherwise, by its siplicity S cetralizes K ad, therefore, G cotais a eleet of order r. By Lea.6, S icludes a Frobeius group F 1 1 with kerel of order q ad cyclic copleet of order q 1. By applyig Lea.5 to the preiage of F i G, we fid that + ω( G); a cotradictio. l 1 + Let S =. Sice OutS, where = log, 16, we have l k ( 1 + log ) 51, ad S 3l. O the other had, S is oe of the groups i Table 1. So, Lea i [9] iplies that S is a siple group of Lie type over a field of characteristic, which cotradicts ε S = D ( q ), where q is odd. () Next, the cases of siple groups of Lie type over fields of characteristic will be cosidered. Sice 16, we have that r, r ( 1), r ( ), ad r ( 3) exist by Lea.3. It iplies that r π( S) by Lea.. Furtherore, r, r ( 1), r ( ), r ( 3) ρ( L) = ρ( G) by Table 8 i [11]. Sice at ost oe prie i ρ ( G) divides K G S, by Propositio 3 i [1], we have r, r( ), r ( 3) π( S), but r ( 1) π( S) or r, r( 1) π( S). If r, r( ), r( 3) π( S), but ( 1) π( S) hold, the we easily get a cotradictio by Lea.3 for S beig oe of A ( q ), A ( q ), r C ( q ), D ( q ), ad D ( q ), where q is eve. Let S A ( ), where = 1 q q = k. We have k = k, ( ) k = ( 1) k, ad ( 3) k = ( ) k, by Lea.3, which are ipossible. We siilarly deal with A ( q ), C ( q ), D ( q ), ad D ( q ), where q is eve. So all of the lead to a cotradictio by Lea.3.

12 68 Next suppose r, r ( 1) π( S). Let S = A 1( q ), where q =. We have k = k ad ( 1) k = ( 1)k by Lea.3. It iplies that = ad k = k. Sice G k = ad ( 1) k S k =, we have ( K G S ) = q. Let S SylK. If K, the G = KNG ( S ) by Frattii s arguet. It follows that S = G K NG ( S ) NG ( S ) K. So r π( NG ( S )) by Lea.. Let x = r ad x NG ( S ). The S < x > is a Frobeius by Lea.. It iplies that r ( S 1) ad S = q, by Lea.3, which is a cotradictio. Now, K = 1 ad G S = q Out( S) = def, where d = (, q 1), e =, f = k. It k follows that k, which is ipossible for 16. We siilarly deal with A ( q ), D ( q ), ad D ( q ), where q is eve. So all of the lead to a cotradictio by Lea.3. Fially, let S C ( q ). We have k = k ad ( 1) k = ( 1)k by Lea.3. It follows that =, k = k, ad S L. It iplies that G L by G = L. This copletes the proof of the theore. Refereces [1] O. A. Alekseeva ad A. S. Kodratiev, Recogizig groups E 8 ( q) by the eleet orders, Ukr. Math. Zh. 5(7) (00), [] O. A. Alekseeva ad A. S. Kodratiev, Quasirecogizability of a class of fiite siple groups by the set of eleet orders, Sib. Math. Zh. () (003), [3] R. Bradl ad W. J. Shi, The characterizatio of PSL (, q) by their eleet orders, Joural of Algebra l63 (199), [] J. H. Coway et al., A Atlas of Fiite Groups, Claredo Press, Oxford, [5] M. A. Grechkoseeva, W. J. Shi ad A. V. Vasil ev, Recogitio by spectru of L 16 ( ), Algebra Colloq. 1() (007), [6] G. Higa, Fiite groups i which every eleet has prie power order, J. Lodo Math. Soc. 3 (1957),

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