COUNTING SUBGROUPS OF NON-EUCLIDEAN CRYSTALLOGRAPHIC GROUPS

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1 MATH. SCAND. 4 (1999), 23^39 COUNTING SUBGROUPS OF NON-EUCLIDEAN CRYSTALLOGRAPHIC GROUPS Abstract GARETH A. JONES A methd is btaied fr cutig the rmal subgrups N f a euclidea crystallgraphic grup withut reflectis, with a give fiite qutiet grup =N; this has applicatis t the eumerati f regular cverigs f rbiflds. The methd, which ivlves M«bius iversi ad character thery, is als applied t cut rmal surface subgrups ad -rmal subgrups f fiite idex i. 1. Itrducti The aim f this te is t describe ad t illustrate a techique fr cmputig the umber G f rmal subgrups N f a -euclidea crystallgraphic grup (r NEC grup), with qutiet grup =N ismrphic t a give fiite grup G. The first part f the methd (due t P. Hall) uses M«bius iversi t reduce the prblem t that f cutig hmmrphisms frm t varius subgrups f G; this ca be applied t ay fiitely geerated grup, ad hece t ay NEC grup. Hwever the secd part, the use f character thery t cut such hmmrphisms, seems t be effective ly fr thse NEC grups which ctai reflectis. Several simple illustrative examples are csidered, where G is cyclic, dihedral, r f prime expet. These methds are extesis f thse used i [9], where is assumed t be a surface grup, that is, it ctais either reflectis r elliptic elemets (see als [10] fr a geeral survey f applicatis f character thery t surfaces). Izquierd has take a cmplemetary apprach i [], where reflectis are allwed, has geus 0, ad G is a dihedral grup D p fr sme prime p; her methds are cmpletely differet, relyig the special structure f these dihedral grups. Oe ca iterpret these eumerative results i terms f rbifld cverigs (see [] fr a geeral accut f this cecti) if is the rbifld h= crrespdig t, where h is the hyperblic plae, the G is the Received August 19, 1996.

2 24 gareth a. jes umber f equivalece classes f regular cverigs f with cverig grup G; the restricti that shuld have reflectis is equivalet t the cditi that the uderlyig surface f shuld be withut budary. By refiig these techiques t cut the rmal surface subgrups f with qutiet-grup G, e ca als cut regular cverigs f by Klei surfaces, rather tha by rbiflds. Similarly, a straightfrward extesi t -rmal subgrups f fiite idex allws e t cut the -regular fiite cverigs f with a give permutati grup as their mdrmy grup. The authr is grateful t the rgaisers ad participats f the EU-fuded Wrkshp Cmputatial Cfrmal Gemetry, Helsiki, 1994, fr valuable discussis which gave rise t this paper, ad t the referee fr sme very helpful cmmets. 2. Cutig rmal subgrups I [5], P. Hall develped a geeral techique fr cmputig the umber G f rmal subgrups N f a fiitely geerated grup with a give fiite qutiet grup =N G. These subgrups N are the kerels f the epimrphisms! G; thesetepi ;G f such epimrphisms is fiite (sice there are ly fiitely may ways f mappig the geeratrs f it G), s G is fiite. If 1 ; 2 2 Epi ;G, theker 1 ˆ ker 2 if ad ly if 2 ˆ 1 fr sme autmrphism f G, s G ˆjEpi ;G =Aut Gj, the umber f rbits f Aut G actig by cmpsiti Epi ;G. Sice this acti is fixed-pit-free, every rbit has legth jaut Gj ad s G ˆjEpi ;G j jaut Gj T cut epimrphisms! G, e first cuts the hmmrphisms, ad the elimiates thse which map t prper subgrups f G. Oeca ivert the equati jhm ;G j ˆ X jepi ;K j ; KG t cut epimrphisms i terms f hmmrphisms, by itrducig the M«bius fucti fr G. This assigs a iteger K t each subgrup K f G by the recursive frmula ( X 1 if K ˆ G; H ˆ K;G ˆ 0 if K < G HK

3 cutig subgrups f -euclidea crystallgraphic grups 25 The equati jepi ;G j ˆ X H jhm ;H j HG is the easily deduced, ad this immediately gives 1 1 X G ˆ H jhm ;H j jaut Gj HG Fr may grups G, itisarutietasktfidjaut Gj ad H fr all H G, s there remais the prblem f cmputig jhm ;H j fr all H G (ratleast,frallthseh G with H 6ˆ0).Weshallusethe character thery f fiite grups t d this fr varius NEC grups ; fr backgrud ifrmati NEC grups ad Klei surfaces, see [1] r [14], fr fiite grups see [7], fr character thery see [7] r [12], ad fr umber thery see [6]. 3. Orietable NEC grups withut reflectis A NEC grup is a discrete grup f ismetries f the hyperblic plae h, with cmpact qutiet space h=. First let us take t be a rietable NEC grup withut reflectis (that is, a c-cmpact Fuchsia grup), s that has sigature S g; ; m 1 ;...; m r Š; f g where g; r; m 1 ;...; m r are itegers with g; r 0adm i > 1fralli. Here g is the geus f h=, the symbl detes rietability, the itegers m i are the perids f, adf g detes the absece f reflectis. This meas that has geeratrs ad defiig relatis X i i ˆ 1;...; r ; A j ; B j j ˆ 1;...; g X m i i ˆ 1 i ˆ 1;...; r ; Y r X i Yg jˆ1 A j ; B j Šˆ1 It fllws that the umber H ˆjHm ;H j f hmmrphisms! H is equal t the umber f slutis x i ; a j ; b j i H f the simultaeus equatis

4 26 gareth a. jes x m i i ˆ 1 i ˆ 1;...; r ; Y r x i Yg jˆ1 a j ; b j Šˆ1 This umber ca be cmputed by meas f the fllwig result, which is prved i [9]. (The case r ˆ 0, where is a rietable surface grup, is due t Frbeius [3] fr g ˆ 1, ad t Medykh [11] fr g > 1. See ½7.2 f [13] fr similar results.) Therem 1. The umber f slutis f equati 2 00 i a fiite grup H, where each x i lies i sme ui L i f cjugacy classes f H, is equal t jhj X 2g 1 X X 1 2 2g r x 1... x r ; x 1 2L 1 x r 2L r where rages ver the irreducible cmplex characters f H. Nte that 1 is the degree f, the dimesi f the crrespdig CHmdule. I rder t deduce a frmula fr H, wewillchsethesetsl i t be the sets f the slutis x i 2 H f equatis 2 0. First we eed sme tati. If m is a psitive iteger, H is a fiite grup ad is a cmplex character f H, thelet H mš ˆfh 2 H j h m ˆ 1 g (which is a ui f cjugacy classes f H), ad let mš ˆ X h The Therem 1 immediately implies h2h mš Crllary 1. If has sigature (S+) ad H is ay fiite grup, the H ˆjHj 2g 1 X 1 2 2g r m 1 Š m r Š ; where rages ver the irreducible cmplex characters f H. Give the character table f H, thisresultmakesitstraightfrwardt evaluate H. Character tables are available fr may fiite grups see [2], fr istace. Befre csiderig sme examples, it is useful t itrduce sme tati we let l dete the least cmm multiple f the perids m 1 ;...; m r f, ad fr each iteger m we let m be the umber f i such that m divides m i.

5 cutig subgrups f -euclidea crystallgraphic grups 27 Examples. (3.1) Let H ˆ C d, a cyclic grup f rder d. I this case, e ca evaluate H directly, withut character thery, fr istace by abeliaisig ad usig the structure thery f fiitely-geerated abelia grups. Hwever, it is useful t apply Crllary 1 here as a simple illustrati f the methd. There are d irreducible characters f H, all f degree 1; these are the hmmrphisms H! C, btaied by mappig a geeratr f H t a d-th rt f 1 i C. Ifm is ay psitive iteger, the H mš is the uique subgrup f rder m; d i H, ads < m; d ; if H mš ker ; mš ˆ 0; therwise. It fllws that a character makes a -zer ctributi t the frmula fr H if ad ly if ker ctais the subgrups H m i Š fr all i ˆ 1;...; r, r equivaletly ker ctais the subgrup H lš f rder l; d which they geerate, where l ˆ lcm m 1 ;...; m r. There are d= l; d such characters, each with m i Šˆ m i ; d, s C d ˆ d2g l; d Yr m i ; d Fr istace, if d is a prime p the < p 2g 1 p ; if p > 0; C p ˆ p 2g ; if p ˆ 0 We ca w cmpute G where G ˆ C, a cyclic grup f rder. First, we have jaut G j ˆ, where is Euler's fucti N. NwG has a uique subgrup H C d fr each divisr d f, ad has ther subgrups; fr each such H we have H ˆ =d, where the right-had side detes the M«bius fucti N. Equati (1) therefre gives C ˆ 1 X dj dj C d d ˆ 1 X d 2g d l; d Yr m i ; d

6 2 gareth a. jes Fr example, if is a prime p the we have < p 2g 1 p 1 = p 1 if p > 0; C p ˆ p 2g 1 = p 1 if p ˆ 0 (3.2) Let H ˆ D p ˆha; b j a p ˆ b 2 ˆ ab 2 ˆ 1i, a dihedral grup f rder 2p, where p is a dd prime. Apart frm the idetity elemet, this grup has e cjugacy class csistig f the p ivlutis a i b,ad p 1 =2classes fa i g f elemets f rder p. Iaddititthepricipalcharacter 1,the ther irreducible characters f H are the alteratig character 2 a i b j ˆ 1 j,tgetherwith p 1 =2characters 0 k k ˆ 1;...; p 1 =2 give by 0 k ai ˆ ik ik ad 0 k ai b ˆ0, where is a primitive pth rt f uity. It fllws that if m; 2p ˆ2p the mš ˆ 2p if ˆ 1; 0 therwise; if m; 2p ˆp the mš ˆ p if ˆ 1 r 2 ; 0 therwise; < p 1 if ˆ 1 ; if m; 2p ˆ2 the mš ˆ 1 p if ˆ 2 ; 2 therwise; if m; 2p ˆ1 the mš ˆ 1 if ˆ 1 r 2 ; 2 therwise. If h detes the umber f i such that m i ; 2p ˆh the Crllary 1 gives D p ˆ 2p 2g 1 2p 2p p p p p p p 1 p 2 p g r 0 2p 0 p ˆ 2p 2g 1 2 2p p p p p p p 1 p 2 p 1 0 p 2 2 2g p 2 whereweiterpret0 0 as meaig 1. If we take G ˆ D p the the subgrups H G are H ˆ D p, a uique subgrup H ˆhai ismrphic t C p, p subgrups ha i bi C 2,adthetrivial subgrup C 1. The values f H fr these subgrups are 1; 1; 1 adp respectively, ad jaut D p j ˆ p p 1, s equati (1) gives 1 D p ˆ D 2p C p p C 2 p C 1 p p 1 ;

7 cutig subgrups f -euclidea crystallgraphic grups 29 We have just cmputed H fr H ˆ D p, ad Example 3.1 deals with the remaiig subgrups H, s e ca substitute these values ad btai D p. As i mst cases where jgj is divisible by mre tha e prime, the resultig geeral frmula is rather uwieldy, s we will mit it ad istead give a simple example. Suppse that has sigature S with m 1 ˆˆm r ˆ p ad r 1; the we fid that H ˆ2 2g p 2g 1 r ; p 2g 1 r ; 2 2g ad 1 fr H ˆ D p ; C p ; C 2 ad 1 respectively, s D ˆ 22g 1 p 2g 2 r 1 p p 1 (This frmula is als valid i the case r ˆ 0,asshwi[9].) (3.3) If H has prime expet p the H mš ˆH r 1 as p des r des t divide m. I the first case the rthgality relatis fr the characters f H give mš ˆjHj r 0 as ˆ 1 (the pricipal character) r 6ˆ 1,adithe secd case each character satisfies mš ˆ 1, the degree f. Itfllws that if p > 0(sthatpjm i fr sme i) the ly 1 ctributes t H, ad we have H ˆjHj 2g 1 p ; if p ˆ 0, the ther had, the m i Šˆ 1 fr all ad all i, sthat H ˆjHj 2g 1 X 1 2 2g (I particular, if H is a elemetary abelia p-grup C p C p,sthat there are jhj characters f degree 1, the < jhj 2g 1 p ; if p > 0; H ˆ jhj 2g ; if p ˆ 0; this geeralises the result i Example 3.1 fr H ˆ C p.) If G has expet p the s des every -trivial subgrup H G, ad e ca apply these frmul, tgether with the values f jaut G j ad H, t determie G. Fr example, let G be the uique -abelia grup f rder p 3 ad expet p (where p > 2). Apart frm G itself, the subgrups H G with H 6ˆ0arethep 1 maximal subgrups, all ismrphic t C p C p ad satisfyig H ˆ 1, tgether with their itersecti (the Frattii subgrup G C p, als equal t the cetre Z G ), fr which H ˆp. It fllws frm the Burside Basis Therem ([7], III.3.15) that a pair f elemets geerate G if ad ly if their images geerate the Frattii

8 30 gareth a. jes factr grup G= G C p C p ; ay tw such pairs are equivalet uder a uique autmrphism f G, s by cutig such pairs we see that Aut G has rder p 3 p p 3 p 2 ˆp 3 p 2 1 p 1. Thus equati (1) becmes 1 G ˆ p 3 p 2 G p 1 C p C p p C p 1 p 1 Sice G has p 2 irreducible characters f degree 1, ad p 1fdegreep, Crllary 1 gives < p 3 2g 1 p ; if p > 0; G ˆ p 3 2g 1 p 2 p 1 p 2 2g ; if p ˆ 0 We have already evaluated H fr the elemetary abelia grups H ˆ C p C p ad C p, s writig 2g 1 p ˆ h we btai < p h 2 p h 1 p h 1 1 = p 2 1 p 1 ; if p > 0; G ˆ p h 1 p h 1 1 p h 1 1 = p 2 1 p 1 ; if p ˆ 0 4. N-rietable NEC grups withut reflectis Nw let be a -rietable NEC grup withut reflectis, s that has sigature S g; ; m 1 ;...; m r Š; f g where g; r; m 1 ;...; m r are itegers with g 1; r 0adm i > 1fralli. Thus has geeratrs ad defiig relatis X i i ˆ 1;...; r ; A j j ˆ 1;...; g X m i i ˆ 1 i ˆ 1;...; r ; Y r X i Yg jˆ1 A 2 j ˆ 1 I this case, the umber H ˆjHm ;H j f hmmrphisms! H is equal t the umber f slutis x i ; a j i H f the simultaeus equatis 3 0 x m i i ˆ 1 i ˆ 1;...; r ;

9 cutig subgrups f -euclidea crystallgraphic grups Y r x i Yg jˆ1 a 2 j ˆ 1 The frmula fr H is similar t that i the rietable case, but it ctais e extra igrediet. Let be a irreducible character f H, affrded by a represetati. TheFrbeius-Schur idicatr f (r f ) is defied t be c ˆ 1 X h 2 jhj We have h2h < 1; if is real, c ˆ 1; if is real but is t real, 0; if is t real. The fllwig result is prved i [9]; the case r ˆ 0 is due t Frbeius ad Schur [4]. Therem 2. The umber f slutis f equati 3 00 i a fiite grup H, where each x i lies i sme ui L i f cjugacy classes f H, is equal t jhj X g 1 c g 1 X X 2 g r x 1... x r ; x 1 2L 1 x r 2L r where rages ver the irreducible cmplex characters f H. Crllary 2. If has sigature (S ) ad H is ay fiite grup, the H ˆjHj g 1 X c g 1 2 g r m 1 Š m r Š ; where rages ver the irreducible cmplex characters f H. Examples. (4.1) Let H ˆ C d, a cyclic grup f rder d. By ur earlier descripti f the characters f H, weseethatc ˆ 0 uless either ˆ 1, the pricipal character give by h ˆ1frallh, relsed is eve ad ˆ 2, the alteratig character which maps a geeratr f H t 1. Bth f these characters have c ˆ 1, s a similar argumet t that i the rietable case gives where C d ˆd g 1 Yg m i ; d

10 32 gareth a. jes < 1; if d/(l,d) isdd, ˆ 2; if d/(l,d) iseve, ad l is the least cmm multiple f the perids m i. Thus if m 2 detes the highest pwer f 2 dividig a iteger m, the >< 1; if m i 2 d 2 fr sme i; ˆ > 2; if m i 2 < d 2 fr all i Fr istace, if we take d t be a dd prime p we btai C p ˆp g 1 p ; whereas by takig d ˆ 2wehave C 2 ˆ 2g 1 2 ; if 2 > 0; 2 g ; if 2 ˆ 0 (4.2) The calculati f D p is similar t that i the rietable case (Example 3.2) the values f mš are uchaged, ad sice every irreducible represetati f a dihedral grup is real, we have c ˆ 1frall. Frm this, e ca deduce a frmula fr D p equati (1) is the same as befre, ad Example 4.1 gives H fr the prper subgrups H f D p. Fr a simple example, suppse that m 1 ˆˆm r ˆ p ad r 1. The H ˆ2 g p g 1 r ; p g 1 r ; 2 g ad 1 fr H ˆ D p ; C p ; C 2 ad C 1 respectively, s D ˆ 2g 1 p g 2 r 1 p p 1 (This frmula is the same as that btaied i the rietable case, except that there each expet g is replaced with 2g.) (4.3) Let H have prime expet p. We ca use the values f mš which we determied i Example 3.3. If p > 0thesicec ˆ 1 fr the pricipal character ˆ 1, we have if p ˆ 0, hwever, the H ˆjHj g 1 p ; H ˆjHj g 1 X c g 1 2 g I this latter case, if p is dd the c ˆ 0frall 6ˆ 1,sthat

11 cutig subgrups f -euclidea crystallgraphic grups 33 H ˆjHj g 1 ˆjHj g 1 p ; as befre, whereas if p ˆ 2thec ˆ 1ad 1 ˆ1 fr every (sice H is abelia), s that H ˆjHj g As i the rietable case, these results exted thse i Example 4.1 fr H ˆ C p. Oe ca w cmpute G whe G has expet p. Fr istace, if G is the -abelia grup f rder p 3 ad expet p (where p > 2, the as befre we have 1 G ˆ p 3 p 2 G p 1 C p C p p C p 1 p 1 Sice p is dd, the abve results give H ˆjHj h fr each subgrup H G the right-had side, where h ˆ g 1 p, s a little algebra yields G ˆph 2 p h 1 p h 1 1 p 2 1 p 1 5. Nrmal surface subgrups Sice ctais reflectis, its trsi elemets are the cjugates f the pwers f the elliptic geeratrs X i. Thus a rmal subgrup N f is trsi-free if ad ly if it ctais -trivial pwers f ay X i,thatis, each X i is mapped t a elemet x i f rder exactly m i i the qutiet-grup G ˆ =N. This is equivalet t N beig a surface grup (rietable r rietable), r equivaletly the rbifld h=n beig a surface (withut cepits). I fact, i this situati h=n is a Klei surface withut budary, its diaalytic structure beig iduced by prjecti frm h; cversely, every cmpact Klei surface withut budary, ther tha the sphere, prjective plae, trus ad Klei bttle, arises i this way (these exceptial surfaces have spherical r euclidea uifrmisatis). Oe ca apply Hall's thery as befre t shw that the umber s G f rmal surface subgrups N i, with =N G, isgiveby 4 s G ˆ 1 X H s jaut Gj H ; HG where s H detes the umber f surface-kerel hmmrphisms! H; these are the hmmrphisms with trsi-free kerel, that is, such that x i ˆ X i has rder m i fr each i ˆ 1;...; r.

12 34 gareth a. jes Fr each fiite grup H ad each iteger m 1, let Hhmi dete the set f elemets f rder m i H, ad fr each character f H let hmi ˆ X h h2hhmi Therems 1 ad 2 immediately imply the tw parts f the fllwig result. Crllary 3. Let H be ay fiite grup. If has sigature (S+) the s H ˆjHj2g 1 X 1 2 2g r hm 1 ihm r i ; ad if has sigature (S ) the s H ˆjHjg 1 X c g 1 2 g r hm 1 ihm r i ; where i each case, rages ver the irreducible cmplex characters f H. Examples. (5.1) Let us calculate s C. All subgrups H f C are cyclic, s first let H ˆ C d, let h be ay geeratr f H, ad let be a primitive d-th rt f 1 i C. The the irreducible characters f H are the hmmrphisms j H! S 1 C, determied by mappig h t j j ˆ 1;...; d ; the kerel f ˆ j has rder k ˆ j; d, ad the image has rder d=k. Ifm des t divide d the hmi ˆ0, s assume that mjd. The Hhmi csists f the elemets f H which are geeratrs f its uique subgrup C m. Nw C m \ ker ˆ C m;k, s maps C m t the grup f m= m; k -th rts f uity, sedig geeratrs t primitive rts; each primitive rt is the image f m = m= m; k elemets f Hhmi, adsice the sum f the primitive -th rts f 1 i C is fr all, wefidthat hmi ˆ m m= m; k m= m; k ; which we will abbreviate t m m m; k (This is the Ramauja sum c m j, thesumfthej-th pwers f the primitive m-th rts f 1, give by c m j ˆ m m m; j ;

13 cutig subgrups f -euclidea crystallgraphic grups 35 where m; k ˆ m; j; d ˆ m; j; d ˆ m; j sice mjd; see [6, ½½ 5.6, 16.6] fr details.) Nw let have sigature (S+). If s C d > 0 the each perid m i must divide d, ad hece their least cmm multiple l must als divide d. Ifwe assume that l divides d, the the abve argumet gives s C d ˆd 2g 1 X Yr 2g 1 ˆ d Yr 2g 1 ˆ d Y r m i X m i X kjd mi m i m i ; k Y r mi m i ; k d Y k r mi ; m i ; k where we have used the fact that there are d=k irreducible characters with j ker j ˆk fr each k dividig d. If we take G ˆ C the all its subgrups H are cyclic grups C d ;weca therefre substitute the abve frmula fr s H i equati (4), givig s C ˆ 1 X dj d s C d ˆ 1 X dd ljdj ˆ 1 Y r m i X ljdj 2g 1 Yr m i X kjd d d X 2g 1 kjd d Y k r d k Y r mi m i ; k mi m i ; k Fr example, let be a prime p. Ifr 1 the the mai sum is empty (ad s s C p ˆ0) uless l ˆ p, thatis,m i ˆ p fr each i. I this case, the ly pssible value fr d i the summati is d ˆ p, sk ˆ 1rp, ad we fid that s C p ˆp 2g 1 p 1 r 1 1 r If r ˆ 0, the ther had, the l ˆ 1, s d ˆ 1rp, adweget s Cp ˆp2g 1 p 1 (This agrees with ur earlier value fr C p,sice is trsi-free whe r ˆ 0.)

14 36 gareth a. jes The calculati f s C d is a little simpler i the -rietable case, sice the ly irreducible characters f C d with c 6ˆ 0arethepricipal character 1 ad the alteratig character 2 (whe d is eve); these tw characters, which sed a geeratr f C d t 1 respectively, satisfy c ˆ 1. As befre, we have hmi ˆ0 uless m divides d, i which case 1 hmi ˆ m ad (if d is eve) 2 hmi ˆ 1 d=m m. Itfllwsthatif! d detes the umber f i such that d=m i is dd, the d g 1 m 1... m r ; if ljd ad d is dd, >< s C d ˆ 2d g 1 m 1... m r ; if ljd, d is eve ad! d is eve, > 0; therwise, s equati (4) gives s C ˆ 1 X ljdj d dd ˆ 1 Y r m i Yr g 1 dd X ljdj d dd m i X ljdj d;! d eve d d g 1 X 2 ljdj d;! d eve Yr g 1 2 dd dd g 1 m i Fr example, let be a dd prime p. Ifr 1the s C p ˆ0 uless l ˆ p, thatis,m i ˆ p fr all i, i which case s C p ˆ p 1 r 1 p g 1 ;ifr ˆ 0 (s that l ˆ 1) the s C p ˆ p g 1 1 = p 1. Similarly, if ˆ p ˆ 2ad r 1weget s C 2 ˆ2 g r 0 as r is eve r dd, while fr r ˆ 0weget s C 2 ˆ2 g 1. As i the rietable case, e ca cfirm these results by csiderig epimrphisms frm the fiite abelia grup = 0 p t C p. (5.2) Let G ˆ D p, where p is a dd prime, ad suppse that has perids m 1 ˆˆm r ˆ p with r 1. A surface-kerel hmmrphism! D p must have image H D p ctaiig elemets x i f rder p, sh ˆ C p r D p ad hece s D 1 p ˆ s p p 1 D p s C p If is rietable, the Example 5.1 shws that s C p ˆp 2g 1 p 1 p 1 r 1 1 r The tw 1-dimesial characters f D p satisfy hpi ˆp 1, while the p 1 =2 remaiig characters (all 2-dimesial) satisfy hpi ˆ 2, s Crllary 3 gives

15 cutig subgrups f -euclidea crystallgraphic grups 37 s D p ˆ 2p 2g 1 X 1 2 2g r hpi r ˆ p 2g 1 p 1 2 2g p 1 r 1 1 r ; frm which we btai s D p ˆp 2g 2 p 1 r 1 2 2g 1 Similarly, i the -rietable case we fid that s C p ˆ p 1 r p g 1 ad s D p ˆ2p g 1 p 1 2 g 1 p 1 r 1 1 r,s s D p ˆp g 2 2 g 1 p 1 r r 6. N-rmal subgrups ad mdrmy grups It is pssible t exted the methd f this paper s that it applies t -rmal subgrups M f fiite idex i a NEC grup, with iducig a give permutati grup their csets. This is because the cre f M (the itersecti f its cjugates i ) is a rmal subgrup N f fiite idex i, ad =N is ismrphic t the fiite trasitive permutati grup G iduced by actig the csets f M. The details are as fllws. Let G be a fiite grup with a faithful, trasitive permutati represetatiaset. Our aim is t determie the umber G; f subgrups M f whse csets iduces a permutati grup =N; =M ismrphic t G;. (A ismrphism betwee permutati grups G; ad G 0 ; 0 csists f a ismrphism G! G 0 ; g 7! g 0 ad a bijecti! 0 ;!7!! 0 such that!g 0 ˆ! 0 g 0 fr all! 2 ad g 2 G.) This exteds ur earlier eumerati f rmal subgrups, crrespdig t the regular represetati f G. Firstly, the techiques f ½½2ö4 are used t fid the umber G f rmal subgrups N f with qutiet-grup =N ismrphic t the abstract grup G. Fr each such cre N, the subgrups M N whse csets iduces a permutati grup ismrphic t G; are i e-t-e crrespdece with the pit-stabilisers G! ˆfg 2 G j!g ˆ!g! 2 i such represetatis f G. Nw these stabilisers frm a rbit uder A ˆ Aut G, s the umber f them is equal t the idex ja N A G! j i A f their rmalisers N A G! ˆf 2 A j G! ˆ G! g, ad hece 5 G; ˆjA N A G! j G The permutati grup G; is ismrphic t the mdrmy grup f the rbifld cverig h=m! ˆ h=, that is, the grup f permutatis f the sheets iduced by liftig clsed paths i. It fllws therefre that

16 3 gareth a. jes this methd eumerates the cverigs f with a give permutati grup as their mdrmy grup. T determie hw may f these cverigs are by Klei surfaces, e restricts this eumerati t surface subgrups M, as i ½5. A subgrup M is a surface grup if ad ly if it ctais elliptic elemets, that is, each f the geeratrs x i i ˆ 1;...; r f G acts as a semi-regular permutati f rder m i, csistig etirely f cycles f legth m i. Oe ca therefre fid the umber s G; f surface subgrups M, whse csets iduces G;, bytakigeachl i i Therem 1 r 2 t be the set f elemets f rder m i i G which act semi-regularly. Example (6.1). Let G ˆ D p actig aturally the set f vertices f aregularp-g, where p is a dd prime. A pit-stabiliser G! C 2 lies i a rbit f legth p uder A ˆ Aut D p, s equati (5) gives D p ; ˆp D p ; the calculati f D p was discussed i Examples 3.2 ad 4.2 i the rietable ad -rietable cases. T determie s D p; e eeds t kw the -idetity elemets f D p which are semiregular ; these are the rtatis a; a 2 ;...; a p 1, which cicide with the elemets f rder p, sifsmem i 6ˆ p the s D p; ˆ0. Let us therefre assume that m i ˆ p fr i ˆ 1;...; r. We may als assume that r 1, fr therwise s D p; ˆ D p ; ad there is prblem. I the rietable case, Example 5.2 shws that there are s D p ˆp 2g 2 p 1 r 1 2 2g 1 pssible kerels N; each is the cre f p surface subgrups M, s s D p; ˆp 2g 1 p 1 r 1 2 2g 1 Similarly, i the -rietable case there are kerels N ad hece there are s D p ˆp g 2 2 g 1 p 1 r r s D p; ˆp g 1 2 g 1 p 1 r r surface subgrups M. I either case, each cjugacy class f these subgrups M csists f all thse with a give cre N, s the umber f cjugacy classes, ad hece the umber f equivalece classes f surface cverigs f with mdrmy grup D p ;, is give by the abve frmul fr s D p.

17 cutig subgrups f -euclidea crystallgraphic grups 39 REFERENCES 1 E. Bujalace, J. J. Etay, J. M. Gamba ad G. Grmadzki, Autmrphism Grups f Cmpact Brdered Klei Surfaces, Lecture Ntes i Math. 1439, J. H. Cway, R. T. Curtis, S. P. Nrt, R. A. Parker ad R. A. Wils, ATLAS f Fiite Grups, Clared Press, Oxfrd, G. Frbeius, U«ber Gruppecharaktere, Sitzber. K«iglich Preuss. Akad. Wiss. Berli (196), 95^ G. Frbeius ad I. Schur, U«ber die reelle Darstelluge der edliche Gruppe, Sitzber. K«iglich Preuss. Akad. Wiss. Berli (1906), 16^20. 5 P. Hall, The Euleria fuctis f a grup, Quarterly J. Math. Oxfrd 7 (1936), 134^ G. H. Hardy ad E. M. Wright, Itrducti t the Thery f Numbers (5th ed.), Oxfrd Uiversity Press, Oxfrd, B. Huppert, Edliche Gruppe I, Spriger^Verlag, Berli / Heidelberg / New Yrk, M. Izquierd, O Klei surfaces ad dihedral grups, Math. Scad. 76 (1995), 221^ G. A. Jes, Eumerati f hmmrphisms ad surface-cverigs, Quarterly J. Math. Oxfrd (2) 46 (1995), 45^ G. A. Jes, Characters ad surfaces a survey, ithe ATLAS te years Prceedigs, Prc. Cf. Birmigham, 1995 (eds. R. T. Curtis ad R. A. Wils), Ld Math. Sc. Lecture Nte Ser., t appear. 11 A. D. Medyh, Determiati f the umber f equivalet cverigs ver a cmpact Riema surface, Dkl. Akad. Nauk SSSR 239 (197), 269^271 (Russia); Sviet Math. Dkl. 19 (197), 31^320 (Eglish traslati). 12 J-P. Serre, Repre setatis liëaires des Grupes fiis, Herma, Paris, J-P. Serre, Tpics i Galis Thery, Jes ad Bartlett, Bst, H. Zieschag, E. Vgt ad H-D. Cldewey, Surfaces ad Plaar Disctiuus Grups, Lecture Ntes i Math. 35, 190. DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTHAMPTON SOUTHAMPTON SO17 1BJ UNITED KINGDOM gaj@maths.st.ac.uk