Polarized non-abelian representations of slim near-polar spaces

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1 Polarzed non-abelan representatons of slm near-polar spaces Bart De Bruyn Bnod Kumar Sahoo Abstract In [15], Shult ntroduced a class of parapolar spaces, the so-called near-polar spaces. We ntroduce here the noton of a polarzed non-abelan representaton of a slm near-polar space, that s, a near-polar space n whch every lne s ncdent wth precsely three ponts. For such a polarzed non-abelan representaton, we study the structure of the correspondng representaton group, enablng us to generalze several of the results obtaned n [14] for non-abelan representatons of slm dense near hexagons. We show that wth every polarzed non-abelan representaton of a slm near-polar space, there s an assocated polarzed projectve embeddng. Keywords. Near-polar space, (unversal, polarzed) non-abelan representaton, (unversal) projectve embeddng, (mnmal) polarzed embeddng, extraspecal -group, combnatoral group theory MSC010: 05B5, 51A45, 51A50, 0F05 1 Introducton Projectve embeddngs of pont-lne geometres have been wdely studed. A projectve embeddng s a map from the pont set of a pont-lne geometry S to the pont set of a projectve space PG(V ) mappng lnes of S to full lnes of PG(V ). In case S has three ponts per lne, the underlyng feld of V s F. For such a geometry, a projectve embeddng can alternatvely be vewed as a map p v p from the pont set of S to the nontrval elements of the addtve group of V such that f {p 1, p, p 3 } s a lne of S, then v p3 = v p1 + v p. Ths alternatve pont of vew allows to generalze the noton of projectve embeddngs to so-called representatons, where ponts of the slm geometry are no longer mapped to ponts of a projectve space or to nonzero vectors of a vector space, but to nvolutons of a group R, the so-called representaton group. If R s a non-abelan group, then the representaton tself s also called non-abelan. Non-abelan representatons have been studed for a varety of geometres, ncludng polar spaces and dense near polygons. In ths paper, we study non-abelan representatons for a class of parapolar spaces that ncludes both the polar spaces and the dense near Supported by DST/SERB (SR/FTP/MS-001/010), Government of Inda 1

2 polygons. Ths class of parapolar spaces was ntroduced by Shult n [15] and called nearpolar spaces n []. In ths paper, we restrct to those near-polar spaces that are slm and to a partcular famly of non-abelan representatons, the so-called polarzed ones. For polarzed nonabelan representatons of slm near-polar spaces, we derve qute some nformaton about the representaton groups. We show that these representaton groups are closely related to extraspecal -groups, and obtan nformaton about the centers of these groups. We also show that wth every polarzed non-abelan representaton of a slm near-polar space, there s an assocated polarzed projectve embeddng (by takng a sutable quotent). Prelmnares.1 Partal lnear spaces and ther projectve embeddngs Let S = (P, L, I) be a pont-lne geometry wth nonempty pont set P, lne set L and ncdence relaton I P L. We call S a partal lnear space f every two dstnct ponts of S are ncdent wth at most one lne. We call S slm f every lne of S s ncdent wth precsely three ponts. In the sequel, all consdered pont-lne geometres wll be partal lnear spaces. We wll often dentfy a lne wth the set of ponts ncdent wth t. The ncdence relaton then corresponds to contanment. A subspace of S s a set X of ponts wth the property that f a lne L has at least two of ts ponts n X then all the ponts of L are n X. A hyperplane of S s a subspace, dstnct from P, meetng each lne of S. The dstance d(x 1, x ) between two ponts x 1 and x of S wll be measured n the collnearty graph of S. A path of mnmal length between two ponts of S s called a geodesc. A subspace X of S s called convex f every pont on a geodesc between two ponts of X s also contaned n X. If x 1 and x are two ponts of S, then the ntersecton of all convex subspaces contanng {x 1, x } s denoted by x 1, x. (Ths s well-defned snce P s a convex subspace.) The set x 1, x tself s a convex subspace and hence t s the smallest convex subspace of S contanng {x 1, x }. The subspace x 1, x s called the convex closure of x 1 and x. A full projectve embeddng of S s a map e from P to the pont set of a projectve space Σ satsfyng: () e(p) Σ = Σ; and () e(l) := {e(x) x L} s a full lne of Σ for every lne L of S. If e s moreover njectve, then the full projectve embeddng e s called fathful. A full projectve embeddng e from S nto a projectve space Σ wll shortly be denoted by e : S Σ. If N s the maxmum dmenson of a projectve space nto whch S s fully embeddable, then the number N + 1 s called the embeddng rank of S and s denoted by er(s). The number er(s) s only defned when S s fully embeddable. Two full projectve embeddngs e 1 : S Σ 1 and e : S Σ of S are called somorphc (denoted by e 1 = e ) f there exsts an somorphsm θ from Σ 1 to Σ such that e = θ e 1. Let e : S Σ be a full projectve embeddng of S and suppose α s a subspace of Σ satsfyng the followng two propertes:

3 (Q1) e(p) α for every pont p of S; (Q) α, e(p 1 ) α, e(p ) for any two dstnct ponts p 1 and p of S. We denote by Σ/α the quotent projectve space whose ponts are those subspaces of Σ that contan α as a hyperplane. Snce α satsfes propertes (Q1) and (Q), t s easly verfed that the map whch assocates wth each pont x of S the pont α, e(x) of Σ/α defnes a full projectve embeddng of S nto Σ/α. We call ths embeddng a quotent of e and denote t by e/α. If S s a fully embeddable slm partal lnear space, then by Ronan [1], S admts up to somorphsm a unque full projectve embeddng ẽ : S Σ such that every full projectve embeddng e of S s somorphc to a quotent of ẽ. The full projectve embeddng ẽ s called the unversal embeddng of S. We have er(s) = dm( Σ) + 1. If S admts a fathful full projectve embeddng, then the unversal embeddng ẽ of S s also fathful.. Near polygons A partal lnear space S = (P, L, I) s called a near polygon f for every pont p and every lne L, there exsts a unque pont on L nearest to p. If d N s the maxmal dstance between two ponts of S (= the dameter of S), then the near polygon s also called a near d-gon. A near 0-gon s a pont, a near -gon s a lne. Near quadrangles are usually called generalzed quadrangles. A near polygon s called dense f every lne s ncdent wth at least three ponts and f every two ponts at dstance have at least two common neghbors..3 Polar and dual polar spaces A partal lnear space S = (P, L, I) s called a polar space f for every pont p and every lne L, ether one or all ponts of L are collnear wth p. The radcal of a polar space s the set of all ponts x whch are collnear wth any other pont. A polar space s called nondegenerate f ts radcal s empty. A subspace of a polar space s sad to be sngular f any two of ts ponts are collnear. The rank r of a nondegenerate polar space s the maxmal length r of a chan S 0 S 1 S r of sngular subspaces where S 0 = and S S +1 for all {0,..., r 1}. A nondegenerate polar space of rank s just a nondegenerate generalzed quadrangle. The rank of a sngular subspace S of a nondegenerate polar space s the maxmal length k of a chan S 0 S 1 S k of sngular subspaces such that S 0 =, S k = S and S S +1 for all {0,..., k 1}. Sngular subspaces of rank r are also called maxmal sngular subspaces, those of rank r 1 are called next-to-maxmal sngular subspaces. A nondegenerate polar space s called thck f every lne s ncdent wth at least three ponts and f every next-to-maxmal sngular subspace s contaned n at least three maxmal sngular subspaces. Wth every (thck) polar space S of rank r 1, there s assocated a partal lnear space, whch s called a (thck) dual polar space of rank r. The ponts of are the maxmal sngular subspaces of S, the lnes of are the next-to-maxmal sngular subspaces of S, 3

4 and ncdence s reverse contanment. Every thck dual polar space of rank r s a dense near r-gon..4 Near-polar spaces In [15], Shult ntroduced a class of pont-lne geometres. These pont-lne geometres were called near-polar spaces n []. Near-polar spaces of dameter n are nductvely defned as follows. A near-polar space of dameter 0 s just a pont and a near-polar space of dameter 1 s a lne havng at least three ponts. A near-polar space of dameter n s a pont-lne geometry S satsfyng the followng fve axoms: (E1) S s connected and ts dameter s equal to n; (E) Every lne of S s ncdent wth at least three ponts; (E3) Every geodesc x 0, x 1,..., x k n S can be completed to a geodesc x 0, x 1,..., x k, x k+1,..., x n of length n; (E4) For every pont x of S, the set H x of ponts of S at dstance at most n 1 from x s a hyperplane of S; (E5) If x 1 and x are two ponts of S wth k := d(x 1, x ) < n, then the subgeometry of S nduced on the convex closure x 1, x s a near-polar space of dameter k. The hyperplane H x mentoned n Axom (E4) s called the sngular hyperplane of S wth deepest pont x. The near-polar spaces of dameter are precsely the nondegenerate polar spaces n whch each lne s ncdent wth at least three ponts. Every near-polar space of dameter n s a strong parapolar space n the sense of Cohen and Coopersten [4]. The convex closures of the pars of ponts at dstance from each other are also called symplecta. Every thck dual polar space and more generally every dense near polygon s a nearpolar space. The class of near-polar spaces also ncludes some half-spn geometres, some Grassmann spaces and some exceptonal geometres, see Shult [15, Secton 6]. We wll now dscuss full projectve embeddngs of near-polar spaces. Most of what we say here s based on De Bruyn [5]. Suppose e : S Σ s a full projectve embeddng of a near-polar space S = (P, L, I). By Shult [15, Lemma 6.1()], every sngular hyperplane H x, x P, of S s a maxmal (proper) subspace. Ths mples that Π x := e(h x ) Σ s ether Σ or a hyperplane of Σ. The embeddng e s called polarzed f Π x s a hyperplane of Σ for every pont x of S. If e s polarzed, then the subspace N e := Π x s called the nucleus of e. By De Bruyn [5, x P Proposton 3.4], the nucleus N e satsfes the condtons (Q1) and (Q) of Secton.1 and the embeddng ē := e/n e s polarzed. Suppose now that S s a slm near-polar space. Then S admts a fathful full polarzed embeddng, see Brouwer & Shpectorov [3] or De Bruyn [5, Proposton 3.11()]. So, S also 4

5 has a unversal embeddng ẽ : S Σ. Ths unversal embeddng necessarly s polarzed and fathful. The embeddng ẽ/nẽ s called the mnmal full polarzed embeddng of S. For every full polarzed embeddng e of S, the embeddng ē = e/n e s somorphc to ẽ/nẽ. Every full embeddng of S s somorphc to ẽ/α for some subspace α of Σ satsfyng Propertes (Q1) and (Q). If α 1 and α are two subspaces of Σ satsfyng (Q1) and (Q), then e/α 1 = e/α f and only f α 1 = α. Suppose agan that S s a slm near-polar space and that e : S Σ s a full polarzed embeddng of S. Ths means that for every pont x of S, the subspace e(h x ) Σ s a hyperplane Π x of Σ. By De Bruyn [5, Propostons 3.5 and 3.11()], the map x Π x defnes a polarzed full embeddng e of S nto a subspace of the dual Σ of Σ. The embeddng e s called the dual embeddng of e. The nucleus of e s empty. So, the dual embeddng e s somorphc to the mnmal full polarzed embeddng of S..5 Extraspecal -groups In the sequel, we wll adopt the followng conventons when dealng wth groups. For elements a, b of a group G, we wrte [a, b] = a 1 b 1 ab and a b = b 1 ab. For elements x, y, z of G, we have [xy, z] = [x, z] y [y, z] and [x, yz] = [x, z][x, y] z. We denote by C n the cyclc group of order n. A fnte -group G s called extraspecal f ts Frattn subgroup Φ(G), commutator subgroup G = [G, G] and center Z(G) concde and have order. We refer to [7, Secton 0, pp.78 79] or [8, Chapter 5, Secton 5] for the propertes of fnte extraspecal -groups whch we wll menton now. An extraspecal -group s of order 1+n for some nteger n 1. Let D 8 and Q 8, respectvely, denote the dhedral and the quaternon groups of order 8. A non-abelan -group of order 8 s extraspecal and s somorphc to ether D 8 or Q 8. If G s an extraspecal -group of order 1+n, n 1, then the exponent of G s 4 and G s ether a central product of n copes of D 8, or a central product of n 1 copes of D 8 and one copy of Q 8. If the former (respectvely, latter) case occurs, then the extraspecal -group s denoted by 1+n + (respectvely, 1+n ). Suppose G s an extraspecal -group of order n+1, n 1, and set G = {1, λ}. Then V = G/G s an elementary abelan -group and hence can be regarded as a n-dmensonal vector space over F. For all x, y G, we defne { 0 f(xg, yg F f [x, y] = 1, ) = 1 F f [x, y] = λ. Then f s a nondegenerate alternatng blnear form on V. For all x G, x G = {1, λ} as G/G s elementary abelan. We defne { 0 q(xg F f x ) = = 1, 1 F f x = λ. Then q s a nondegenerate quadratc form on V. The blnear form assocated wth q s precsely f, that s, q(xg yg ) = q(xg ) + q(yg ) + f(xg, yg ) 5

6 for all x, y G. The nondegenerate quadratc form q defnes a nonsngular quadrc of PG(V ), whch s of hyperbolc type f G = 1+n + or of ellptc type f G = 1+n..6 Representatons of slm partal lnear spaces Let S = (P, L, I) be a slm partal lnear space. A representaton [10, p.55] of S s a par (R, ψ), where R s a group and ψ s a mappng from P to the set of nvolutons n R, satsfyng: () R s generated by the mage of ψ; () ψ(x)ψ(y) = ψ(z) for every lne {x, y, z} of S. If {x, y, z} s a lne of S, then condton () mples that ψ(x), ψ(y), ψ(z) are mutually dstnct and [ψ(x), ψ(y)] = [ψ(x), ψ(z)] = [ψ(y), ψ(z)] = 1. The group R s called a representaton group of S. The representaton (R, ψ) of S s called fathful f ψ s njectve. Dependng on whether R s abelan or not, the representaton (R, ψ) tself wll be called abelan or non-abelan. For an abelan representaton, the representaton group s an elementary abelan -group and hence can be consdered as a vector space over the feld F wth two elements. In ths case, the representaton thus corresponds to a full projectve embeddng of S. We refer to [9] and [13, Sectons 1 and ] for representatons of partal lnear spaces wth p + 1 ponts per lne, where p s a prme. Suppose S 1 and S are two slm partal lnear spaces. Let (R, ψ ), {1, }, be a representaton of S. The representatons (R 1, ψ 1 ) and (R, ψ ) are called equvalent f there exsts an somorphsm θ 1 from S 1 to S and a group somorphsm θ from R 1 to R such that ψ θ 1 (x) = θ ψ 1 (x) for every pont x of S 1. If S 1 = S, then (R 1, ψ 1 ) and (R, ψ ) are called somorphc f there exsts a group somorphsm θ from R 1 to R such that ψ (x) = θ ψ 1 (x) for every pont x of S 1. Suppose (R, ψ) s a representaton of a slm partal lnear space S. Let N be a normal subgroup of R such that ψ(x) N for every pont x of S. For every pont x of S, let ψ N (x) denote the element ψ(x)n of the quotent group R/N. Then (R/N, ψ N ) s a representaton of S whch s called a quotent of (R, ψ). If (R 1, ψ 1 ) and (R, ψ ) are two representatons of S, then (R, ψ ) s somorphc to a quotent of (R 1, ψ 1 ) f and only f there exsts a group epmorphsm θ from R 1 to R such that ψ (x) = θ ψ 1 (x). If ths s the case, then (R, ψ ) s somorphc to (R 1 /N, (ψ 1 ) N ), where N = ker(θ)..7 Polarzed and unversal representatons of slm near-polar spaces Let S = (P, L, I) be a slm near-polar space of dameter n. A representaton (R, ψ) of S s called quas-polarzed f [ψ(x), ψ(y)] = 1 for every two ponts x and y of S at dstance at most n 1 from each other. 6

7 An abelan representaton (R, ψ) of S s called polarzed f the correspondng full projectve embeddng (n the sense of Secton.6) s polarzed. A non-abelan representaton (R, ψ) of S s called polarzed f [ψ(x), ψ(y)] = 1 for every two ponts x and y of S at dstance at most n 1 from each other, that s, f the representaton s quas-polarzed. We wll later show that wth every polarzed non-abelan representaton of S, there s an assocated full polarzed embeddng of S (whch s obtaned by takng a sutable quotent). (1) Let R u be the group defned by the generators r x, x P, and the followng relatons: r x = 1, where x P; r x r y r z = 1, where x, y, z P such that {x, y, z} L. For every pont x of S, we defne ψ u (x) := r x R u. () Let R p be the group defned by the generators r x, x P, and the followng relatons: r x = 1, where x P; [r x, r y ] = 1, where x, y P such that d(x, y) < n; r x r y r z = 1, where x, y, z P such that {x, y, z} L. For every pont x of S, we defne ψ p (x) := r x R p. (3) As mentoned before, S has fathful full projectve embeddngs. The unversal projectve embeddng of S can be constructed as follows. Let V be a vector space over F wth a bass B whose elements are ndexed by the ponts of P, say B = { v x x P}. Let W be the subspace of V generated by all vectors v x1 + v x + v x3 where {x 1, x, x 3 } s some lne of S. Let Ṽ be the quotent vector space V/W and for every pont x of S, let ṽ x be the vector v x + W of Ṽ. The map x ṽ x defnes a full projectve embeddng ẽ of S nto PG(Ṽ ) whch s somorphc to the unversal embeddng of S. Proposton.1. (1) ( R u, ψ u ) s a fathful representaton of S. () ( R p, ψ p ) s a fathful polarzed representaton of S. (3) If (R, ψ) s a representaton of S, then (R, ψ) s somorphc to a quotent of ( R u, ψ u ). (4) If (R, ψ) s a quas-polarzed representaton of S, then (R, ψ) s somorphc to a quotent of ( R p, ψ p ). Proof. We show that ( R p, ψ p ) s a fathful representaton. Snce ṽ x + ṽ x = W for every x P, ( ṽ x ) + ( ṽ y ) + ṽ x + ṽ y = W for all x, y P and ṽ x + ṽ y + ṽ z = W for every lne {x, y, z} of S, we know from von Dyck s theorem that there exsts an epmorphsm from R p to the addtve group of Ṽ mappng r x to ṽ x for every pont x of S. Snce ẽ s a 7

8 full projectve embeddng, ṽ x W and hence r x Rp 1 for every x P. The latter fact mples that ( R p, ψ p ) s a representaton. Snce ẽ s a fathful projectve embeddng, we have ṽ x ṽ y for any two dstnct ponts x, y P. Ths mples that also r x Rp r y. So, ( R p, ψ p ) s a fathful representaton. In a completely smlar way, one can show that ( R u, ψ u ) s a fathful representaton. Clams (3) and (4) are straghtforward consequences of von Dyck s theorem. By constructon, the representaton ( R p, ψ p ) s quas-polarzed and hence polarzed f R p s non-abelan. Suppose R p s abelan. Then let e p denote the full projectve embeddng of S correspondng to ( R p, ψ p ). Let ( R, ψ) denote the abelan representaton correspondng to the unversal projectve embeddng ẽ of S. By Clam (4), ( R, ψ) s somorphc to a quotent of ( R p, ψ p ), and hence ẽ s somorphc to a quotent of e p. As ẽ cannot be a proper quotent of some full embeddng of S, the projectve embeddngs ẽ and e p are somorphc. So, e p s polarzed, or equvalently, ( R p, ψ p ) s polarzed. The representaton ( R u, ψ u ) s called the unversal representaton of S. The representaton ( R p, ψ p ) s called the unversal polarzed representaton of S. From Secton 5 (see Lemma 5.3) t wll follow that there exsts a λ R p such that [ ψ p (x), ψ p (y)] = λ for every two ponts x and y at dstance n from each other. If λ = 1, then the unversal polarzed representaton s abelan and hence corresponds to the unversal projectve embeddng of S (whch s always polarzed). If λ 1, then the unversal polarzed representaton of S s non-abelan. Both nstances can occur. Indeed, the slm dual polar space DW (n 1, ) and the slm dense near hexagons Q(5, ) L 3, Q(5, ) Q(5, ) have non-abelan polarzed representatons [6, 11], whle no fnte slm nondegenerate polar space has non-abelan representatons [13, Theorem 1.5()]. Computer computatons showed that other dense near polygons (lke the dual polar space DH(5, 4)) also have non-abelan polarzed representatons (n extraspecal -groups), but the authors are stll lookng for computer free descrptons of these representatons. 3 Man results For a fnte slm near-polar space S, we denote the embeddng rank er(s) also by er + (S). The vector space dmenson of the mnmal full polarzed embeddng of S wll be denoted by er (S). We wll see n Proposton 4. that the number er (S) s even. By [14], every non-abelan representaton of a slm dense near hexagon s polarzed. The followng theorem s the frst man theorem of ths paper. It generalzes some results regardng slm dense near hexagons obtaned n [14]. We wll prove t n Secton 5. Theorem 3.1. Suppose S s a fnte slm near-polar space of dameter n havng some polarzed non-abelan representaton (R, ψ). Then n 3 and the unversal polarzed representaton ( R p, ψ p ) of S s also non-abelan. Moreover, () ψ s fathful and ψ(x) / Z(R) for every pont x of S. 8

9 () R s a -group of exponent 4, R = and R = Φ(R) Z(R). () If Z(R) = l+1, then Z(R) s somorphc 1 to ether (C ) l+1 or (C ) l 1 C 4. (v) R s of order β for some nteger β satsfyng 1+er (S) β 1+er + (S). We have β = 1 + er (S) f and only f R s an extraspecal -group. We have β = 1 + er + (S) f and only f (R, ψ) s somorphc to ( R p, ψ p ). (v) If l = er + (S) er (S), then Z( R p ) has order l+1 and so s somorphc to ether (C ) l+1 or (C ) l 1 C 4. In Secton 6, we prove the followng results. Theorem 3.. Suppose S s a fnte slm near-polar space of dameter n 3 havng polarzed non-abelan representatons. Then the followng hold: () The polarzed representatons of S are precsely the representatons of the form ( R p /N, ( ψ p ) N ), where N s a subgroup of R p contaned n Z( R p ). () If N 1 and N are two subgroups of Z( R p ), then the representatons ( R p /N 1, ( ψ p ) N1 ) and ( R p /N, ( ψ p ) N ) of S are somorphc f and only f N 1 = N. Remark. If l = er + (S) er (S), then we wll see n Secton 6 that Theorems 3.1(v) and 3. mply that the number of nonsomorphc polarzed non-abelan representatons of S s equal to the sum l+1 [ ] l+1 l [ ] l f Z( R p ) = (C ) l+1, and equal to l [ ] l l 1 [ ] l 1 =0 f Z( R p ) = (C ) l 1 C 4. =0 Theorem 3.3. Suppose S s a fnte slm near-polar space of dameter n 3 havng polarzed non-abelan representatons. Set l := er + (S) er (S). Then S has a polarzed non-abelan representaton (R, ψ) wth R = 1+er (S) f and only f Z( R p ) = (C ) l+1. If ths s the case then there are up to somorphsm l such representatons. Moreover, the representaton groups of any two of them are somorphc (to ether 1+er (S) + or 1+er (S) ). Theorem 3.4. Suppose S s a fnte slm near-polar space of dameter n 3 havng polarzed non-abelan representatons. Suppose Z( R p ) = C l 1 C 4, where l = er + (S) er (S) 1. Then R +er (S) for every polarzed non-abelan representaton (R, ψ) of S. Moreover, there are up to somorphsm l 1 polarzed non-abelan representatons (R, ψ) wth R = +er (S). If (R, ψ) s such a representaton, then Z(R) = C 4. 1 If l = 0, then (C ) 1 s not defned. In ths case, ths sentence should be understood as Z(R) s somorphc to C. The terms occurrng n ths sum are Gaussan bnomal coeffcents. =0 =0 9

10 4 Some propertes of near-polar spaces Let S be a near-polar space of dameter n 1. Two ponts x and y of S are called opposte f they are at a maxmum dstance from each other, that s, d(x, y) = n. For two dstnct ponts x, y of S, we wrte x y f they are collnear. Proposton 4.1. Let S be a near-polar space of dameter n 1. Let Γ be the graph whose vertces are the ordered pars of opposte ponts of S, wth two dstnct vertces (x 1, y 1 ) and (x, y ) beng adjacent whenever ether x 1 = x and y 1 y ; or x 1 x and y 1 = y. Then Γ s connected. Proof. Let (x 1, y 1 ) and (x, y ) be two arbtrary vertces of Γ. We prove that (x 1, y 1 ) and (x, y ) are contaned n the same connected component of Γ. For every pont x of S, the subgraph of the collnearty graph of S nduced on the set of ponts at dstance n from x s connected by Shult [15, Lemma 6.1()]. So, f x 1 = x or y 1 = y, then (x 1, y 1 ) and (x, y ) belong to the same connected component of Γ. Assume that x 1 x and y 1 y. We prove that there exsts a pont y 3 at dstance n from x 1 and x. If y 3 s such a pont, then (a 1, b 1 ) and (a, b ) belong to the same connected component of Γ for every (a 1, b 1, a, b ) {(x 1, y 1, x 1, y 3 ), (x 1, y 3, x, y 3 ), (x, y 3, x, y )}, provng that (x 1, y 1 ) and (x, y ) also belong to the same connected component of Γ. The pont y 3 alluded to n the prevous paragraph s defned as a pont of S at dstance n from x 1 whch les as far away from x as possble. Suppose d(y 3, x ) n 1 for such a pont y 3. Then by Axom (E3), there exsts a pont y 4 collnear wth y 3 whch les at dstance k := d(y 3, x ) + 1 from x. By Axom (E5), a near-polar space of dameter k can be defned on the convex closure x, y 4. By applyng Axom (E4) to ths near-polar space of dameter k, we see that the ponts of the lne y 3 y 4 dstnct from y 3 le at dstance k = d(y 3, x )+1 from x. By Axoms (E) and (E4) appled to S, at least one of the ponts of y 3 y 4 \ {y 3 } les at dstance n from x 1. Ths contradcts the maxmalty of d(y 3, x ). So, d(x 1, y 3 ) = d(x, y 3 ) = n as we needed to prove. Proposton 4.. Let S = (P, L, I) be a fnte slm near-polar space of dameter n 1, let V be a fnte-dmensonal vector space over F and let e : S PG(V ) be a full polarzed embeddng of S nto PG(V ). Then there exsts a unque alternatng blnear form f on V for whch the followng holds: If x s a pont of S and v s the unque vector of V for whch e(x) = v, then PG( v f ) s a hyperplane of PG(V ) whch contans all the ponts e(y), where y P and d(x, y) n 1, and none of the ponts e(z), where z P and d(x, z) = n. If e s somorphc to the mnmal full polarzed embeddng of S, then the alternatng blnear form f s nondegenerate and hence er (S) = dm(v ) s even. Proof. For every pont x of S, let Π x denote the unque hyperplane of PG(V ) whch contans all the ponts e(y), where y P and d(x, y) n 1, and none of the ponts e(z), where z P and d(x, z) = n. 10

11 (1) We frst prove the exstence of the alternatng blnear form n the case e s somorphc to the mnmal full polarzed embeddng of S. Then Π x =. Recall that the map x Π x defnes a full projectve embeddng e of S nto the dual PG(V ) of PG(V ). Ths embeddng e s called the dual embeddng of e and s somorphc to the mnmal full polarzed embeddng of S. So, there exsts an somorphsm φ from PG(V ) to PG(V ) mappng e(x) to Π x for every pont x of S. We prove that φ s a polarty of PG(V ), or equvalently that φ = 1. Snce φ defnes a collneaton of PG(V ), t suffces to prove that φ (p) = p for every pont p belongng to a generatng set of PG(V ). So, t suffces to prove that φ(π x ) = φ (e(x)) = e(x) for every pont x of P. If y s a pont at dstance at most n 1 from x, then e(y) Π x mples that φ(π x ) Π y. Hence, φ(π x ) s contaned n the ntersecton I of all hyperplanes Π y, where y P and d(x, y) n 1. Snce e s polarzed, the hyperplanes Π y, where y P and d(x, y) n 1, generate a hyperplane of PG(V ). So, I s a sngleton. Snce e(x) Π y for every y P satsfyng d(x, y) n 1, we also have e(x) I. Hence, φ(π x ) = e(x) as we needed to prove. We now prove that φ s a symplectc polarty of PG(V ). To that end, t suffces to prove that p p φ for every pont p of PG(V ). Snce PG(V ) = Im(e), t suffces to prove the followng: (a) e(x) e(x) φ for every x P; (b) f L = {p 1, p, p 3 } s a lne of PG(V ) such that p 1 p φ 1 and p p φ, then also p 3 p φ 3. Snce e(x) φ = Π x and e(x) Π x, Property (a) clearly holds. If p p φ 1, then {p 3 } L p φ 1 p φ = L φ p φ 3. If p p φ 1, then p φ 1 = L φ, p 1, p φ = L φ, p and p φ 3 s the unque hyperplane through L φ dstnct from p φ 1 and p φ, mplyng that p φ 3 = L φ, p 3. So, Property (b) also holds n that case. If f s the nondegenerate alternatng blnear form of V correspondng to the symplectc polarty φ of PG(V ), then f satsfes the requred condtons. () Suppose e s not somorphc to the mnmal full polarzed embeddng of S. Let α be the ntersecton of all subspaces Π x, x P, let U be the subspace of V correspondng to α and let W be a subspace of V such that V = U W. For every pont x of S, let e (x) denote the unque pont of PG(W ) contaned n α, e(x). Then e s somorphc to the mnmal full polarzed embeddng of S. By part (1) above, we know that there exsts a nondegenerate alternatng blnear form f W on W such that f x s a pont of S and w s the unque vector of W for whch e (x) = w, then the hyperplane PG( w f W ) of PG(W ) contans all ponts e (y), where y P and d(x, y) n 1, and none of the ponts e(z), where z P and d(x, z) = n. Now, for all ū 1, ū U and all w 1, w W, we defne x P f(ū 1 + w 1, ū + w ) := f W ( w 1, w ). Then f s an alternatng blnear form on V. Suppose x s a pont of S. Let v be the unque vector of V for whch e(x) = v and let w be the unque vector of W for whch e (x) = w. Then w = U, v W. We 11

12 also have v f = U, w fw. Snce PG( w f W ) contans all ponts e (y), where y P and d(x, y) n 1, and none of the ponts e (z), where z P and d(x, z) = n, we have that PG( v f ) contans all ponts e(y), where y P and d(x, y) n 1, and none of the ponts e(z), where z P and d(x, z) = n. So, the alternatng blnear form f satsfes the requred condtons. (3) We now prove the unqueness of the alternatng blnear form. Suppose f 1 and f are two alternatng blnear forms on V satsfyng the requred condtons. Then g := f 1 f s also an alternatng blnear form on V. Suppose x 1 and x are two ponts of S and let v, {1, }, be the unque vector of V for whch e(x) = v. If d(x 1, x ) n 1, then f 1 ( v 1, v ) = 0 = f ( v 1, v ) and hence g( v 1, v ) = 0. If d(x 1, x ) = n, then f 1 ( v 1, v ) = 1 = f ( v 1, v ) and hence g( v 1, v ) = 0. Snce PG(V ) = e(x) x P, we get g = 0. Hence f 1 = f. 5 Structure of the representaton groups Let S = (P, L, I) be a fnte slm near-polar space of dameter n and suppose (R, ψ) s a polarzed non-abelan representaton of S. In ths secton, we wll prove all the clams mentoned n Theorem 3.1. Lemma 5.1. We have n 3. Proof. By [13, Theorem 1.5()], every representaton of a fnte slm nondegenerate polar space s abelan. So, S s not a polar space and hence n 3. Lemma 5.. The unversal polarzed representaton ( R p, ψ p ) s non-abelan. Moreover, R p 1+er+ (S). Proof. As (R, ψ) s a quotent of ( R p, ψ p ), the unversal polarzed representaton ( R p, ψ p ) tself should also be non-abelan. Snce the abelan representaton correspondng to the unversal projectve embeddng of S s quas-polarzed, t should be a quotent of ( R p, ψ p ) by Proposton.1(4). Ths mples that R p 1+er+ (S). Later (Lemma 5.1) we wll show that R p = 1+er+ (S). Lemma 5.3. Let Γ be the graph as defned n Proposton 4.1. Then there exsts an nvoluton λ R such that λ = [ψ(x), ψ(y)] for every vertex (x, y) of Γ. Proof. We frst show that [ψ(x 1 ), ψ(y 1 )] = [ψ(x ), ψ(y )] for any two adjacent vertces (x 1, y 1 ) and (x, y ) of Γ. Suppose x 1 = x and y 1 y. Let y 3 be the unque thrd pont of the lne y 1 y. Then d(x 1, y 3 ) = n 1. Snce ψ(y 3 ) commutes wth ψ(x 1 ) and ψ(y ), we have [ψ(x 1 ), ψ(y 1 )] = [ψ(x 1 ), ψ(y )ψ(y 3 )] = [ψ(x 1 ), ψ(y )]. The case where x 1 x and y 1 = y s treated n a smlar way. Now let x and y be two opposte ponts of S and set λ = [ψ(x), ψ(y)]. By Proposton 4.1, Γ s connected. So, by the frst paragraph, λ s ndependent of the opposte ponts x and y. Also λ 1 snce (R, ψ) s polarzed and non-abelan. Snce λ 1 = [ψ(x), ψ(y)] 1 = [ψ(y), ψ(x)] = λ, we get λ = 1. 1

13 Corollary 5.4. ψ(x), ψ(y) = D 8 for every two opposte ponts x and y of S. Proof. Snce x and y are opposte ponts, (ψ(x)ψ(y)) = [ψ(x), ψ(y)] = λ by Lemma 5.3 and so ψ(x)ψ(y) s of order 4. Hence ψ(x), ψ(y) = D 8 [1, 45.1]. Lemma 5.5. ψ s fathful and ψ(x) / Z(R) for every pont x of S. Proof. Let x and y be two dstnct ponts of S and let z be a pont that s opposte to x, but not to y (such a pont exsts by Axom (E3)). Then [ψ(y), ψ(z)] = 1 and [ψ(x), ψ(z)] = λ 1 by Lemma 5.3. Hence, ψ(x) ψ(y). For a gven pont x, choose a pont w opposte to x. Then [ψ(x), ψ(w)] = λ 1. So ψ(x) / Z(R). Lemma 5.6. R = {1, λ} Z(R). Proof. Set T = λ = {1, λ}. Then T R by Lemma 5.3. We frst show that T Z(R). Snce R = ψ(x) x P, t s suffcent to prove that [ψ(x), λ] = 1 for every pont x of S. Let y be a pont of S opposte to x. Snce ψ(x), ψ(y), λ = [ψ(x), ψ(y)] all are nvolutons, a drect calculaton shows that [ψ(x), λ] = 1. Beng a central subgroup, T s normal n R. In the quotent group R/T, the generators ψ(x)t, x P, commute parwse. So R/T s abelan and hence R T. Corollary 5.7. For a, b, c R, [ab, c] = [a, c][b, c] and [a, bc] = [a, b][a, c]. Proof. By Lemma 5.6, we have [ab, c] = [a, c] b [b, c] = [a, c] [b, c] and [a, bc] = [a, c] [a, b] c = [a, b] [a, c]. Lemma 5.8. (1) For every r R, we have r {1, λ}. () R s a fnte -group of exponent 4 and R = Φ(R). Proof. We show that r {1, λ} for every r R\{1}. Set r = ψ(x 1 )ψ(x ) ψ(x n ), where x 1, x,..., x n are ponts of S. Snce λ = 1, ψ(x ) = 1 and [ψ(x ), ψ(x j )] {1, λ} Z(R) for all, j {1,..., n}, we have r {1, λ}. It follows that r 4 = 1. Snce R s non-abelan, the exponent of R cannot be and hence equals 4. Snce R = ψ(x) x P and S s fnte, the quotent group R/R = ψ(x)r x P s a fnte elementary abelan -group. Snce R = by Lemma 5.6, we get that R s also a fnte -group. Then the two facts that R s the smallest normal subgroup K of R such that R/K s abelan and that Φ(R) s the smallest normal subgroup H of R such that R/H s elementary abelan [1, 3., p.105] mply R = Φ(R). Snce the quotent group R/R s an elementary abelan -group, we can consder V = R/R as a vector space over F. For every pont x of S, let e(x) be the projectve pont ψ(x)r of PG(V ). Notce that, by Lemmas 5.5 and 5.6, ψ(x)r s ndeed a nonzero vector of V. Lemma 5.9. The map e defnes a fathful full projectve embeddng of S nto PG(V ). 13

14 Proof. Snce R/R = ψ(x)r x P, the mage of e generates PG(V ). We prove that ψ(x 1 )R ψ(x )R for every two dstnct ponts x 1 and x of S. Suppose to the contrary that ψ(x 1 )R = ψ(x )R. Snce ψ s fathful by Lemma 5.5, we have ψ(x 1 ) ψ(x ). So, ψ(x 1 ) = ψ(x )λ. By Axom (E3), there exsts a pont x 3 opposte to x 1, but not to x. Then λ = [ψ(x 1 ), ψ(x 3 )] = [ψ(x )λ, ψ(x 3 )] = [ψ(x ), ψ(x 3 )] = 1, a contradcton. Let L = {x 1, x, x 3 } be a lne of S. We have e(x ) = ψ(x )R, for {1,, 3}. Snce ψ(x 1 )ψ(x ) = ψ(x 3 ), we have ψ(x 3 )R = ψ(x 1 )R ψ(x )R. Hence {e(x 1 ), e(x ), e(x 3 )} s a lne of PG(V ). Defnton. For all a, b R, we defne f(ar, br ) = { 1 f [a, b] = λ, 0 f [a, b] = 1. Snce R = {1, λ} Z(R), the map f : V V F s well-defned. Lemma The map f : V V F s an alternatng blnear form of V. Proof. The clam that f s an alternatng blnear form follows from the followng facts. Snce [a, a] = [1, a] = [a, 1] = 1, we have f(ar, ar ) = f(r, ar ) = f(ar, R ) = 0 for all a R. Let x 1, x, y 1 R. Snce [x 1 x, y 1 ] = [x 1, y 1 ][x, y 1 ], we have f(x 1 R x R, y 1 R ) = f(x 1 R, y 1 R ) + f(x R, y 1 R ). Let x 1, y 1, y R. Snce [x 1, y 1 y ] = [x 1, y 1 ][x 1, y ], we have f(x 1 R, y 1 R y R ) = f(x 1 R, y 1 R ) + f(x 1 R, y R ). Lemma The embeddng e of S nto PG(V ) s polarzed. Proof. For every pont x of S, we defne a certan subspace Π x of PG(V ). Let v be the unque vector of V for whch e(x) = v. Then Π x s the subspace of PG(V ) correspondng 3 to the subspace v f of V. Let x 1 and x be two ponts of S and let v = ψ(x )R, {1, }. So e(x ) = v. Then the followng holds: d(x 1, x ) n 1 [ψ(x 1 ), ψ(x )] = 1 f(ψ(x 1 )R, ψ(x )R ) = 0 f( v 1, v ) = 0 v v f 1 e(x ) Π x1. Now from the above t follows that Π x = e(h x ) PG(V ) s a hyperplane of PG(V ) for every pont x of S, where H x s the sngular hyperplane of S wth deepest pont x. So e s polarzed. 3 The map φ x : R R defned by φ x (r) = [ψ(x), r] s a homomorphsm (see Corollary 5.7) whch s surjectve. The kernel of φ x s C R (ψ(x)) whch has ndex n R by the frst somorphsm theorem. Then v f s precsely the mage of C R (ψ(x)) n V under the canoncal homomorphsm R V ; r rr. 14

15 Defnton. We call e the full polarzed embeddng of S assocated wth the non-abelan representaton (R, ψ). Lemma 5.1. (1) R s of order β for some β satsfyng 1 + er (S) β 1 + er + (S). () The followng are equvalent: (R, ψ) s somorphc to ( R p, ψ p ); β = 1 + er + (S); e s somorphc to the unversal embeddng of S. (3) The followng are equvalent: β = 1 + er (S); R s an extraspecal -group; e s somorphc to the mnmal full polarzed embeddng of S. Proof. By Lemmas 5.9 and 5.11, e defnes a full polarzed embeddng of S nto PG(V ). So, er (S) dm(v ) er + (S). Snce R/R = β 1, we have dm(v ) = β 1 and hence 1 + er (S) β 1 + er + (S). The lower bound occurs f and only f e s somorphc to the mnmal full polarzed embeddng of S. The upper bound occurs f and only f e s somorphc to the unversal embeddng of S. From Lemma 5., the upper bound and the fact that (R, ψ) s somorphc to a quotent of ( R p, ψ p ), t follows that β = 1 + er + (S) f and only f (R, ψ) s somorphc to ( R p, ψ p ). Now, R s extraspecal f and only f R = Z(R), that s, f and only f the alternatng blnear form f s nondegenerate. For every pont x of S, let v x be the unque vector of V for whch e(x) = v x. Then e(h x ) = PG( v x f ) (see the proof of Lemma 5.11) s a hyperplane of PG(V ) for every pont x of S. It follows that f s nondegenerate f and only f the nucleus N e of e s empty, that s, f and only f e s a mnmal full polarzed embeddng of S. Thus R s extraspecal f and only f er (S) = dm(v ) = β 1. For every r R, we set θ(r) := rr V. Observe that f r 1, r R, then f(θ(r 1 ), θ(r )) = 0 f [r 1, r ] = 1 and f(θ(r 1 ), θ(r )) = 1 f [r 1, r ] = λ. We denote by R f the radcal of the alternatng blnear form f. The subspace of PG(V ) correspondng to R f s precsely N e. Lemma If N s a subgroup of R contaned n Z(R), then θ(n) R f. Proof. Let g N and h R. Then [g, h] = 1 mples that f(θ(g), θ(h)) = 0. Snce θ(r) = V, t follows that θ(g) R f. Hence, θ(n) R f. Lemma If U s a subspace of R f, then θ 1 (U) s a subgroup of R contaned n Z(R). If dm(u) = l, then θ 1 (U) s an abelan subgroup somorphc to ether C l+1 or C l 1 C 4. 15

16 Proof. Clearly, θ 1 (U) s a subgroup of R. If g θ 1 (U) and h R, then we have f(θ(g), θ(h)) = 0 snce θ(g) U R f. Ths mples that [g, h] = 1. So, θ 1 (U) Z(R). In partcular, θ 1 (U) s abelan. By the classfcaton of fnte abelan groups, θ 1 (U) s somorphc to the drect product of a number of cyclc groups. Snce the exponent of R s equal to 4, each of these cyclc groups has order or 4. Lemma 5.8(1) then mples that there s at most one cyclc group of order 4 n ths drect product. If dm(u) = l, then θ 1 (U) = l+1 and hence θ 1 (U) must be somorphc to ether (C ) l+1 or (C ) l 1 C 4. Corollary (1) We have R f = θ(z(r)). () We have Z(R) = R er (S). (3) If l = er + (S) er (S), then the center Z( R p ) of R p s somorphc to ether C l+1 or C l 1 C 4. Proof. (1) By Lemmas 5.13 and 5.14, we have θ(z(r)) R f and θ 1 (R f ) Z(R), mplyng that R f = θ(z(r)). () Snce λ Z(R), we have Z(R) = θ(z(r)) = R f = V er (S) = R er (S). (3) Ths follows from Lemma 5.14 and Clam (). We wll now study the quotent representatons of (R, ψ). For such quotent representatons, we need normal subgroups N of R such that ψ(x) N for every pont x of S. Lemma The normal subgroups of R are the followng: (1) the subgroups of R contanng λ; () the subgroups of R not contanng λ that are contaned n Z(R). Proof. Clearly, the subgroups n (1) and () above are normal n R. Suppose N s a normal subgroup of R not contanng λ. For all n N and all r R, we then have [n, r] N R = N {1, λ} = {1}, mplyng that N Z(R). Remark. If N s a (normal) subgroup of R contaned n Z(R), then the condton that ψ(x) N for every pont x of S s automatcally satsfed by Lemma 5.5. Lemma Let N be a normal subgroup of R such that ψ(x) N for every pont x of S. Then the quotent representaton (R/N, ψ N ) s abelan f and only f λ N. Proof. The representaton (R/N, ψ N ) s abelan f and only f [ψ(x)n, ψ(y)n] = [ψ(x), ψ(y)] N = N for every two ponts x and y of S, that s, f and only f λ N. Lemma Let N be a (necessarly normal) subgroup of R contanng λ such that ψ(x) N for every pont x of S. Set U := θ(n), and let α denote the subspace of PG(V ) correspondng to U. Then e(x) α for every pont x of S, and the full projectve embeddng of S correspondng to the abelan representaton (R/N, ψ N ) s somorphc to e/α. 16

17 Proof. If x P, then the facts that ψ(x) N, λ N and R = {1, λ} mply that ψ(x)r does not belong to the set {nr n N}, that s, e(x) α. So, e/α s well-defned as a projectve embeddng. Consder the quotent vector space V/U and the assocated projectve space PG(V/U). The map whch sends each pont x of S to the pont (ψ(x)r ) U of PG(V/U) s then a full projectve embeddng somorphc to e/α. The map φ : R/N V/U; rn θ(r) U (r R), whch s well-defned as U = θ(n), s an somorphsm of groups. (The njectvty of the map follows from the fact that θ 1 (U) = N whch s a consequence of the fact that λ N.) The fact that φ ψ N (x) = φ(ψ(x) N) = θ(ψ(x)) U = (ψ(x)r ) U for every pont x of S mples that the full projectve embeddng of S correspondng to the abelan representaton (R/N, ψ N ) s somorphc to e/α. Lemma Let N be a subgroup of R contaned n Z(R). Set U := θ(n) R f, and let α N e denote the subspace of PG(V ) correspondng to U. Then: (1) If λ N, then the projectve embeddng assocated wth the non-abelan representaton (R/N, ψ N ) s somorphc to e/α. () The representaton (R/N, ψ N ) s polarzed. Proof. (1) Consder the normal subgroup N := N, λ Z(R) of R. By Lemma 5.5, ths group does not contan any element ψ(x) where x P. We have θ(n) = θ(n) = U. So, by Lemma 5.18, the projectve embeddng correspondng to the abelan representaton (R/N, ψ N ) s somorphc to e α where α s the subspace of PG(V ) correspondng to U. It s straghtforward to verfy that the projectve embeddng assocated wth the non-abelan representaton (R/N, ψ N ) s somorphc to the projectve embeddng correspondng to the abelan representaton (R/N, ψ N ). (Observe that R/N = (R/N)/(N/N) and (R/N) = N/N.) () If (R/N, ψ N ) s non-abelan, then the fact that [ψ(x)n, ψ(y)n] = [ψ(x), ψ(y)] N = N for any two non-opposte ponts x and y mples that (R/N, ψ N ) s polarzed. If (R/N, ψ N ) s abelan, then the fact that α N e mples that e/α s polarzed and hence that (R/N, ψ N ) s polarzed by Lemma Lemma 5.0. Let N be a normal subgroup of R such that ψ(x) N for every pont x of S. Then the representaton (R/N, ψ N ) s polarzed f and only f N Z(R). Proof. If N Z(R), then (R/N, ψ N ) s polarzed by Lemma Conversely, suppose that (R/N, ψ N ) s polarzed. If λ N, then N Z(R) by Lemma So, we may suppose that {1, λ} N. Then (R/N, ψ N ) s an abelan representaton of S. Now, R/N = (R/R )/(N/R ), where R = {1, λ}. The embeddng e has PG(V ) as target projectve space, where V = R/R s regarded as an F -vector space. The full projectve embeddng e correspondng to (R/N, ψ N ) has PG(R/N) as target projectve space, where the elementary abelan -group R/N s agan regarded as an F -vector space. Snce e s polarzed, we should have θ(n) = N/R R f, that s, N Z(R). 17

18 6 Classfcaton of the polarzed non-abelan representatons In ths secton, we shall prove all the clams mentoned n Theorems 3., 3.3 and 3.4. Let S = (P, L, I) be a fnte slm near-polar space of dameter n 3 that has polarzed non-abelan representatons. We set ( R, ψ) equal to ( R p, ψ p ), the unversal polarzed representaton of S. There then exsts an element λ R \ {1} such that [ ψ(x), ψ(y)] = λ for every two opposte ponts x and y of S. Recall also that R = {1, λ} and that the quotent group R/ R s an elementary abelan -group whch can be regarded as a vector space Ṽ over F. Let f denote the alternatng blnear form on Ṽ assocated wth ( R, ψ) as descrbed n Secton 5 (see Lemma 5.10). The radcal of f s denoted by R f. For every r R, we put θ(r) := r R Ṽ and for every pont x of S, we put ẽ(x) equal to the pont ψ(x) R of PG(Ṽ ). Then ẽ s somorphc to the unversal embeddng of S. By Secton 5, we also know the followng. Proposton 6.1. The polarzed representatons of S are precsely the representatons of the form ( R/N, ψ N ), where N s a subgroup contaned n Z( R). Recall that f N s a subgroup contaned n Z( R), then N necessarly s normal and ψ(x) Z( R) for every pont x of S, mplyng that the quotent representaton ( R/N, ψ N ) s well-defned. Proposton 6.. If N 1 and N are two subgroups of R contaned n Z( R), then the quotent representatons ( R/N 1, ψ N1 ) and ( R/N, ψ N ) of S are somorphc f and only f N 1 = N. Proof. We prove that f the representatons ( R/N 1, ψ N1 ) and ( R/N, ψ N ) are somorphc, then N 1 N. By symmetry, we then also have that N N 1. Let φ be a group somorphsm from R/N 1 to R/N such that φ( ψ(x)n 1 ) = ψ(x)n for every pont x of S. Let g N 1. Snce R = ψ(x) x P, there exst (not necessarly dstnct) ponts x 1, x,..., x k such that g = ψ(x 1 ) ψ(x ) ψ(x k ). Then N = φ(n 1 ) = φ(gn 1 ) = φ( ψ(x 1 )N 1 ψ(x k )N 1 ) = φ( ψ(x 1 )N 1 ) φ( ψ(x k )N 1 ) = ψ(x 1 )N ψ(x k )N = gn. Hence, g N. Snce g s an arbtrary element of N 1, we have N 1 N. By Corollary 5.15(3), we know that Z( R) s somorphc to ether (C ) l+1 or (C ) l 1 C 4, where l := er + (S) er (S). Proposton 6.3. () The number of nonsomorphc polarzed representatons of S s equal to the sum l+1 [ ] l+1 f Z( R) l [ ] = (C ) l+1, and equal to l l 1 [ ] l 1 f =0 l 1 and Z( R) = (C ) l 1 C 4. =0 =0 18

19 () The number of nonsomorphc polarzed non-abelan representatons of S s equal to l+1 [ ] l+1 l [ ] l f Z( R) = (C ) l+1, and equal to l [ ] l l 1 [ ] l 1 f l 1 and =0 =0 Z( R) = (C ) l 1 C 4. Proof. By Lemma 5.17 and Propostons 6.1 and 6., the number of nonsomorphc polarzed (non-abelan) representatons of S s equal to the number of subgroups of Z( R) (not contanng λ). If Z( R) = (C ) l+1, then Z( R) s an elementary abelan -group and so the number of subgroups of Z( R) (contanng λ) s equal to l+1 [ ] ( l l+1 [ ] ) l. If Z( R) = (C ) l 1 C 4, then Z( R)/ λ = (C ) l and hence the total number of subgroups of Z( R) contanng λ s equal to l [ ] l. If G s a subgroup of Z( R) not contanng =0 λ, then G only has elements of order 1 and. The subgroup of Z( R) consstng of all elements of order 1 and s somorphc to (C ) l and hence the number of subgroups of Z( R) not contanng λ s s equal to l [ ] l l 1 [ ] l 1. =0 Lemma 6.4. The followng are equvalent: (1) Z( R) s elementary abelan, that s, somorphc to C l+1 ; () S has a non-abelan representaton (R, ψ), where R s some extraspecal group; (3) S has a non-abelan representaton (R, ψ), where R = 1+er (S). If one of these condtons hold, then the number of nonsomorphc polarzed non-abelan representatons (R, ψ) wth R = 1+er (S) s equal to l. Proof. In Lemma 5.1(3), we already showed that () and (3) are equvalent. By Lemma 5.17 and Proposton 6.1, S has polarzed non-abelan representatons (R, ψ) where R = 1+er (S) f and only f Z( R) has subgroups of order l not contanng λ. Such subgroups do not exst f l 1 and Z( R) = (C ) l 1 C 4. If Z( R) = (C ) l+1, then the number of such subgroups s equal to [ l+1 l ] [ l l 1 ] =0 =0 = l. Lemma 6.5. If l 1 and Z( R) = (C ) l 1 C 4, then R +er (S) for every polarzed non-abelan representaton (R, ψ) of S. The number of such polarzed non-abelan representatons (up to somorphsm) s equal to l 1. If (R, ψ) s a polarzed non-abelan representaton of S for whch R = +er (S), then Z(R) = C 4. Proof. By Lemmas 5.1 and 6.4, we know that R +er (S) for every polarzed nonabelan representaton (R, ψ) of S. The number of such polarzed non-abelan representatons (up to somorphsm) s equal to the number of subgroups of order l 1 of Z( R) that do not contan λ, that s, equal to [ l l 1 ] [ l 1 l 19 ] =0 =0 =0 = l 1. Suppose (R, ψ) s a polarzed

20 non-abelan representaton of S wth R = +er (S). Then Z(R) s somorphc to ether C 4 or C C by Lemma 5.14 and Corollary 5.15(). If Z(R) = C C, then Z(R) contans subgroups of order not contanng R and so (R, ψ) has a proper quotent whch s a polarzed non-abelan representaton. Ths s mpossble as the sze of the representaton group R s already as small as possble. Lemma 6.6. If N 1 and N are two subgroups of R contaned n Z( R) such that λ N 1 N and θ(n 1 ) = θ(n ), then there exsts an automorphsm of R mappng N 1 to N. As a consequence, the quotent groups R/N 1 and R/N are somorphc. Proof. Set U := θ(n 1 ) = θ(n ) = v 1, v,..., v k for some vectors v 1, v,..., v k of Ṽ where k = dm(u). Put d := dm(ṽ ) and extend { v 1, v,..., v k } to a bass { v 1, v,..., v d } of Ṽ. For every {1,,..., d}, let g be an arbtrary element of θ 1 ( v ). For all, j {1,,..., d}, put a j := 1 f f( v, v j ) = 1 and a j := 0 otherwse. The group R has order d+1 and conssts of all elements of the form λ ɛ 0 g ɛ 1 1 g ɛ g ɛ d d, where ɛ 0, ɛ 1,..., ɛ d {0, 1}. If, j {1,,..., d}, we have [g, g j ] = 1 f f( v, v j ) = 0 and [g, g j ] = λ f f( v, v j ) = 1. So, the multplcaton nsde the group R should be as follows. If ɛ 0, ɛ 1,..., ɛ d, ɛ 0, ɛ 1,..., ɛ d {0, 1}, then ( λ ɛ 0 g ɛ 1 1 g ɛ g ɛ d d ) ( λ ɛ 0 g ɛ 1 1 g ɛ g ɛ d d ) = λ ɛ 0+ɛ 0 +ɛ 0 g ɛ 1 +ɛ 1 1 g ɛ +ɛ g ɛ d+ɛ d d, where ɛ 0 := d d =1 j=+1 a jɛ ɛ j. Recall that λ N 1 N. So, for every {1,,..., k}, there exsts a unque element g (1) {g, g λ} belongng to N1 and a unque element g () {g, g λ} belongng to N. Then N 1 = g (1) 1, g (1),..., g (1) k and N = g () 1, g (),..., g () k. Now, let I denote the subset of {1,,..., k} consstng of all {1,,..., k} for whch g (1) g (), or equvalently, for whch g () = g (1) λ. Then the permutaton of R defned by λ ɛ 0 g ɛ 1 1 g ɛ g ɛ d d λ ɛ 0+ɛ 0 g ɛ 1 1 g ɛ g ɛ d d, where ɛ 0 := I ɛ, s an automorphsm φ of R. Snce φ(g (1) ) = g () {1,,..., k}, we have φ(n 1 ) = N. for every Corollary 6.7. If (R 1, ψ 1 ) and (R, ψ ) are two polarzed non-abelan representatons of S for whch the assocated full polarzed embeddngs are somorphc, then also the representaton groups R 1 and R are somorphc. Proof. Let N 1 and N be the subgroups of R contaned n Z( R) such that (R 1, ψ 1 ) = ( R/N 1, ψ N1 ) and (R, ψ ) = ( R/N, ψ N ). Then λ N 1 N. Let α 1 and α be the subspaces of Nẽ correspondng to, respectvely, U 1 := θ(n 1 ) R f and U := θ(n ) R f. By Lemma 5.19(1), the projectve embeddngs e/α 1 and e/α are somorphc. Ths mples that α 1 = α. Hence, θ(n 1 ) = θ(n ). By Lemma 6.6, R 1 = R. 0

21 Proposton 6.8. If (R 1, ψ 1 ) and (R, ψ ) are two polarzed non-abelan representatons of S such that R 1 = R = β, where β = 1 + er (S), then R 1 and R are somorphc (to ether β + or β ). Proof. Let N 1 and N be the unque normal subgroups of R contaned n Z( R) such that λ N 1 N and ( R/N 1, ψ N1 ) = (R 1, ψ 1 ) and ( R/N, ψ N ) = (R, ψ ). Then N 1 = N = R R 1 = l, where l = er + (S) er (S). Snce λ N 1 N and Z( R) = l+1, we have Z( R) = N 1, λ = N, λ. Hence, R f = θ(z( R)) = θ( N 1, λ ) = θ(n 1 ) = θ( N, λ ) = θ(n ). By Lemma 6.6, R 1 = R/N1 = R/N = R. References [1] M. Aschbacher. Fnte group theory. Cambrdge Studes n Advanced Mathematcs 10, Cambrdge Unversty Press, Cambrdge, 000. [] R. J. Blok, I. Cardnal, B. De Bruyn and A. Pasn. Polarzed and homogeneous embeddngs of dual polar spaces. J. Algebrac Combn. 30 (009), [3] A. E. Brouwer and S. V. Shpectorov. Dmensons of embeddngs of near polygons. Unpublshed manuscrpt. [4] A. M. Cohen and B. N. Coopersten. A characterzaton of some geometres of Le type. Geom. Dedcata 15 (1983), [5] B. De Bruyn. Dual embeddngs of dense near polygons. Ars Combn. 103 (01), [6] B. De Bruyn, B. K. Sahoo and N. S. N. Sastry. Non-abelan representatons of the slm dense near hexagons on 81 and 43 ponts. J. Algebrac Combn. 33 (011), [7] K. Doerk and T. Hawkes. Fnte soluble groups. de Gruyter Expostons n Mathematcs 4, Walter de Gruyter & Co., Berln, 199. [8] D. Gorensten. Fnte groups. Chelsea Publshng Co., New York, [9] A. A. Ivanov. Non-abelan representatons of geometres. Groups and Combnatorcs n memory of Mcho Suzuk, , Adv. Stud. Pure Math. 3, Math. Soc. Japan, Tokyo, 001. [10] A. A. Ivanov, D. V. Pasechnk and S. V. Shpectorov. Non-abelan representatons of some sporadc geometres. J. Algebra 181 (1996), [11] K. L. Patra and B. K. Sahoo. A non-abelan representaton of the dual polar space DQ(n, ). Innov. Incdence Geom. 9 (009),