Abstract. x )-elationj and. Australasian Journal of Combinatorics 3(1991) pp

Transcription

1 Abstract x )-elatinj and f this paper frm part f a PhD thesis submitted the authr t the "",,,,,,.,.,,,,,t,, f Lndn. The authr the supprt f the Cmmnwealth :::;c.hjlarsrup Cmmissin. Australasian Jurnal f Cmbinatrics 3(1991) pp

2 term (a) D n and (b) NIH ~ q) fr sme s, 0 pwer q. s::=;n

3 rder. f "'-CleSll~n D* translatin de!icrllpt:ln f this We shall need the... v... vv... u.j<;. result f 237

4 Therem Let D* be a D(n,q) with n 3 and q 2, let T* be a set f translatin blcks f D*, and let S* = nyet* y. that S* is nt t and des nt cntain the intersectin f any tw blcks f D* and that every blck f D* S* is a translatin blck. Then either D* r D*(S*) is D(s,q) fr sme s, 2 s ~ n - 3, and D* is,,,,,...-n,r.1'.,,,hll'" t btained frm by substituting D*(S*) fr the design f pints hv1npt'nll"n~>'" f an s-dimensinal f Here D*(S*) dentes the design n S* whse blcks are the distinct sets x n S* as x ranges ver the set f all blcks f D* which d nt cntain S*. Finally we shall need the tllvltlnu pwerful therem f M. E. O'Nan. ~~~~ (O'Nan [10]) "J.H~VJ.,(;.uJ.. t()geth<~r with related results and C01[lst,rucuns, is discussed in [12], We assume, thrughut, the I,.IUld.1CM:::S f Therem B. In this sectin we cnstruct frm (D, 2-design Dl and N ~ ), and then prceed t (using the therem f sme m ~ 2 and q, rv In analgy with the cnstructin f qutient spaces frm n..."""",...,,,"" spaces, we define the Dl f as fllws. The are just the elements f T as subsets f the f f Dl are the distinct as X ranges ver the whle f in is just set inclusin. When we wish t that an element y f T is cnsidered as a in (rather than blck f D) we shall dente it [yj. Similarly, when we wish t emphasise that the set X' (cnsisting f varius pints f is being cnsidered as a blck in we shall dente it by [X']. The sets X' are cnstant size j. Prf The grup N fixes S pintwise, is transitive n S (Result 1.8), and maps T nt itself. 0 Lemma 2.2 Dl is a 2-(:Q, If,~) design with Q = blcks, :Q = ITI pints, r. k-j l blcks n a pint and ~ = blcks n tw pints. 238

5 lv 18 uanshlve n ", eacn OlOCK la < J InClaent WHn tne same numer. Nte that the blcks f f hence N is transitive n T with cnstant number r: f blcks f L,-CleSlllln since tw In rder aut,mlrl>hu)ffis f inherited frm we define a [yo]; K:N [(BO)']. au1~nl0rph]lsm f (c) nt fix fllws since K, is the blcks f laftl 1 r p. Since a des, hence it has rder p. Dente by IN([Y]) the under K, f IN(Y). Since IN(Y) is an elementary p-grup, s is IN([Y])' The next lemma determines the f NK-. 239

6 in 'Y N'" with exactly 2-transitive n the f OUIJ";J..UUV f Prf x E T, Fllws frm Lemma 2.4 since there Ex. (b) As N is transitive n T, Nit is transitive n the in. We shw that transitive n the relnaining [y], [z] be any tw distinct f, distinct frm If [x], [y] and are all S [w] nt culne,a,r and [x], [z] and [w] are nt cllinear. A repeat f the abve argument required aultil[loi'phlsil[l. ( c) fllws frm the fact that in N'Jj' a grup homc~m4[)n:,hi:!iin and IN(x) nrmal G* be a sult>l.u:ouln Aut(D*). that there Y and blcks x, y with x, X tj. y and Y E y, such that y) x) are nn-trivial. Then la* x) and y) are abelian grups. This uses tec:hniqtles and {J E define the maps fa and is abelian. {J f-'; {J] a f-'; HVU-J.U'I;J.ULl'J elements f Let 'Y = {J] fr sme Result 1.3., E definitin, s a2'y = (J-la2{J a~, an elatin with centre Xf3 ::J. a2'y are elatins with distinct centres and hence cmmute (a10:2), = 0:1(a2,) = (01.2,)0:1 = (0:2ad'Y' Hence 0:10:2 = a20:1 as re~uired. EIa" (X, x) is a grup, it is an abelian grup. {J] T shw that EIa" (Y, y) is an abelian grup, use the dual f the result fr E1a" (X, x). prved Fr each z E T, IN(z) is an elementary abelian p-grup. 240

7 there nn-trivial. that n~r-np'l"n',"'n'p. Dl which x). Then a lt frm thse f in the 241

8 If fllws frm the definitin "blcks" in Dl that the h~t n""1fn!...",,, (4)rreSl)OIldlng t tw distinct blcks f must themselves be distinct. Hwever as many blcks pints and hv1rlpl nl:>t.nl j;! D(r) crresdcmds it has +1, +1, IJ'VId.U;;~C" f Therem B. In this sectin we first shw blck in a translatin blck. We then n1"ru'p/ rj f Therem q. Fr any nulmbers m, D: + k (h) A - lsi = jq(m _. 1). + palt"an:j.eters f are 'Q exl)ressi()ns fr 'Q, b., v and k -1 in k) - 1) and s frm we btain exl)ressic)ns fr A, v k, k A (j-=--q_~_l + lsi) (jq=) = Simplifying this exl>re!lsicm ISI= which prves 242

9 I" It q 1) + 1. ::SUbstItutmg and 0 and Y E t... ""... "':;t:;.,,,'gk n T. Suppse and is transitive n S, d(y) where 1 is in t'ln(x) lsi) (1).LI"LJLU.AJ'''' =J. S S AB and D I 1, Ie are a cnstant size. then blck size Prf As T we have Lemma i=i (1)

10 Cunt the tlags y) where P E :.J and y E '1', 1:.JIVe 11."1) = Lemma (2) Cunt the (P, Q,y) where Q E P -I- Q and y E ISI(ISI- 1)('\ IT!) = i=l 1) frm ISI(ISI (q + 1) + q) = ISI(ISI We shw that = (ISI- fr i = 1,..., Cnsider i=l -2 S, frm (2) and (1) 2 1) + =0 Since the left hand side is a sum f squares, it fllws that each term must be zer Le. l'pil IS~-!. fr i = 1,..., lsi. 0 Every blck f D which cntains S t T. Prf If lsi> 1 and yet then, by Lemma y meets S in fewer than lsi pints and s y des nt cntain S. If lsi = 1 then N has nly tw blck rbits: ne must cnsists f the blcks which cntain S and the ther f the blcks which d nt cntain S. 244

11 ,i lsi. (b) Fr all :c E T, N z is transitive n each This is since if lsi> 1, lsil = q Nw cnsider the actin f IN(X) n acts n Ai. (dual f JU... uu." f IAil. Hwever is transitive n Yl Y2 T with Xl rf- YI and t, Yl) and then via Y2) where E Y2 \:c. s YI c;. :c U Y2 and E Yl \(x U Y2) and = pd(y), fixes the (k - YET\{:e} 1) + 1) (1) Recall the grup holmomc,rpjtns:m N ~ N K.. This results in the Nit ~ PSL(m + 1, q) s l liv([x])1 = under K, f IN(x)). If I ker(k,)n ln(x)1 = Fr Y E T\{x}, 245

12 and s pd(y) /pz = qm-l, petty) fr all y E T\ {x}. S pd(y) is qm-lpz, sinceker(,.;)n IN(z) = ker(,.;)n( IN(x)y) cnstant pd., say, fr all y E T\{x}. Let yet. We shw that lyeln (z) I ::; q. Fr if> E IN ( X ), yrp is a blck nt equal t x and cntaining x n y. S distinct elements f yeln(z) intersect exactly in x n y. There are at mst v-ie blcks in yeln(z) that is Ie-A, lyeln(z)1 ::; v - k = q. jqm jqm-l by Lemma 3.1 the rbit-stabilizer therem, I IN(X)yllyelN(z)1 = I IN(x)l. As lyeln(:il)i ::; q and lyein(:il)iii ln(x)i, we have lyeln(:il) I \ q. Hence I IN(x)1 I ln(x)yl qm-lpzp/(y») _ (v -ITI) = (ITI- l)(pzqm-l - 1) + qm-lpz I)pfCY) 1) + qm-lpz(v IT!) - (v - ITI) -1) Hence ((k - l)q - (v 1)) (k -1) + (v 1) = qm-lpz L(pf(Y) - 1) Hwever (k - l)q - (v - 1) = (k - l)q - tk(k - 1). We nw shw that k-;.-l = q. Using Lemma 3.1 we have k = jq(m) + f=t and hence k - 1 = jq(m) + Again by Lemma 3.1, ). = jq(m - 1) + ;=t and s k 1 j(q I)Q(m) + (j - q) j(q - I)Q(m - 1) + (j 1) yet 246

13 1) 1) (2) We shw J L grup in N" with Dn~-lIll1all~e H the restrictin f C011ntm2 the elements f L Hence v - blck. k translatin n, and s j = Lemma 3.1 Q( m + t + 1), and "~"'U"".UhJ.J...J D is a symmetric "'-Ul~::>lg.u. since m 2:

14 Prf f 'Therem tl tly Lemmas t).q,.1.1 anu.:>.0,.ij HI a. U"", 'l} 1.U1. "V.HJ.~ n ~ 3 and pwer q, every blck in T is a translatin blck, and every blck f D which cntains S belngs t T. Als, since CD, G) is in Class B, each pint f S lies n at least tw blcks in and therefre S is nt equal t and des nt cntain the intersectin f any tw blcks f D. It by Results 1.9 and 1.6, that either (i) D P n,q r (ii) D2 = D(S) is a D(s, q) fr sme s, 2 ::; s ::; n - 3, and D is ismrphic t a design D* btained frm P n,q by substituting the design fr the design f pints and hyperplanes f suitable s-dimensinal OUIU"IJa,,'CO U f P n,q' Suppse that (ii) hlds. Then the pint-set f D (strictly D*) is btained frm that f by replacing the pint-set f U by the pint-set f D 2 ; and, if y is a blck f P n,q then the blck y' in D is btained by replacing y n U by (y n U)B if y jj U (r by S if y :J U), where (J is the fixed bijectin frm the blck-set f U t the blck-set f D2 (as in [8], p239). Nw cnsider the map 17 frm D t defined as fllws. Each blck z f D is mapped 7] t the (unique) blck y f P n,q such that z y'. Each pint f D which lies in S is mapped by 7] t itself. If XES then there is a nn-identity elatin a in N with centre X and axis x, ::) S. Nw a in an bvius way, an autmrphism f3 f which fixes x pintwise but has n further fixed pints. Since f3 is an axial autmrphism f Pn,q, it has a (unique) centre Y. If y' is any blck f D which cntains then a fixes y' and s f3 fixes y, that is y cntains Y. S the blcks y f such that y' :3 X have a unique cmmn pint Y. Y E U. Fr X E we define 7](X) t be the pint Y cnstructed in this way. But then 7] is an ismrphism frm D t P n,q (as it maps cncurrent blcks t cncurrent blcks). S D P n,q' By Therem N / H PSL(m+ 1, q). But n ~ m+ 1 by the prf f Lemma 3.8, and m ~ 2. S N / H ~ PSL(n - 5, q) fr sme 5, 0 ::; 5 S n - 3. This cmpletes the prf f Therem B. 0 The pairs (D, G) in Class C are examined in [5], where under extra cnditins it is shwn that D is a D(n, q) and that D has a subspace isrmrphic t P 8,q fr sme 8,2::; 5 ::; n - 2. In particular, if 5 = n - 2 then D Pn,q r D is btained frm P n,q by a prcess called K -alteratin. I am very grateful t the referee wh has given me excellent advice n the rerganisatin and rewriting f the paper. I wuld als like t thank the referee fr his/her cntributins t the prfs f Lemma 3.6 and Lemma

15 References 1. Andre, J., 'Uber Persektivitaten in endlichen prjektiven Ebenen', Arch. Math. 6 (1954) Butler, N. T., 'On semi-translatin blcks', Gem. Dedicata 16 (1984) Butler, N. 'Symmetric designs f types V and VI', J. Stat. Plan. and In/. 11 (1985) Jacksn, W. 'On the elatin structure f specific designs', t appear in Gem. Dedicata 5. Jacksn, W. 'Elatins and symmetric prjective subspaces', t appear in Gem. Dedicata 6. Kantr, W. 'Elatins f designs', Ganad. J. Math. 5 (1970) Kelly, G. S, On Autmrphisms f Symmetric 2-designs (PhD The3is) University f Lndn (1979) 8. Kelly, G. S., 'Symmetric designs with translatin blcks, I', Gem. Dedicata 15 {1984} Kelly, G. S., 'Symmetric designs with translatin blcks, II', Gem. Dedicata 15 (1984) O'Nan, M. 'Nrmal structure f the ne-pint stabilizer f a dublytransitive permutatin grup.!', Trans. Amer. Math. Sc. 214 (1975) Piper, F. C., 'Elatins f finite prjective planes', Math. Z. 82 (1963) Tsuzuku, T., Finite Grups and Finite Gemetries Cambridge University Press (1982) 13. Wagner, A., ton perspectivities f finite prjective planes', Math. Z. 11 (1959)

16