On cyclotomic schemes over finite near-fields

DOI 10.1007/s10801-007-0081-4

On cyclotomic schemes over finite near-fields

J. Bagherian·Ilia Ponomarenko· A. Rahnamai Barghi

Received: 24 March 2006 / Accepted: 22 May 2007 / Published online: 16 June 2007

© Springer Science+Business Media, LLC 2007

Abstract We introduce a concept of cyclotomic association scheme over a finite near-field K. It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V ), whereV is the linear space associated withK. A sufficient condition on a cyclotomic schemeCthat guarantee the inclusion Aut(C)≤AL(1,F),whereFis a finite field with|K|elements, is given.

Keywords Association scheme·Finite near-field·Permutation group

1 Introduction

An algebraic structureK=(K,+,◦)is called a (right) near-field ifK⁺=(K,+)is a group with the neutral element 0_K,K^×=(K\ {0_K},◦)is a group,x◦0_K=0_Kfor allx∈K,and

(x+y)◦z=x◦z+y◦z, x, y, z∈K. (1) In the finite case, the groupK⁺is elementary Abelian, and the groupK^×is Abelian iffKis a field (as to near-fields theory, we refer to [13]). By the Zassenhaus theorem

I. Ponomarenko partially supported by RFFI, grants 03-01-00349, NSH-2251.2003.1.

J. Bagherian·A. Rahnamai Barghi (

Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45195-1159, Zanjan, Iran e-mail: [email protected]

J. Bagherian

e-mail: [email protected] I. Ponomarenko

Petersburg Department of V.A. Steklov Institute of Mathematics, Fontanka 27, St. Petersburg 191023, Russia

e-mail: [email protected]

apart from seven exceptional cases, each finite near-fieldKis the Dickson near-field, i.e., there exist a finite fieldF0and its extensionFsuch thatF⁺=K⁺and

y◦x=y^σx·x, x, y∈K, (2) whereσ_x∈Aut(F/F0)and·denotes the multiplication inF. In this case,|F0| =q and|K| = |F| =qⁿ, whereqis a power of a certain primep,andn= [F:F0]. It can be proved that(q, n)forms a Dickson pair, i.e., every prime factor ofnis a divisor of q−1 and 4|nimplies 4|(q−1). There exist exactlyϕ(n)/ knonisomorphic Dickson near-fields corresponding to the same Dickson pair(q, n),wherek is the order of p (modn). The multiplicative group of any Dickson near-field is solvable (and even meta-cyclic).

Let K be a finite near-field and K be a subgroup of the group K^×. Set R= {R_a}a∈K, where

Ra=

(x, y)∈K²: y−x∈a◦K

. (3)

Then it is easily seen that any element ofRis a 2-orbit of the permutation group Γ (K,K)= {x→x◦b+c, x∈K: b∈K, c∈K}, (4) and so the pair(K,R)forms an association scheme onK(see Sect.2for the back- ground on permutation groups and association schemes). We call it the cyclotomic scheme over the near-fieldKand denote it by Cyc(K,K). The number|K|is called the valency of the scheme. IfK=K^×, then the scheme is of rank 2, and we call it the trivial scheme. The set of all cyclotomic schemes of valencym < qⁿ−1 over a Dickson near-field corresponding to a Dickson pair(q, n)is denoted by Cyc(q, n, m).

WhenK=Fis a field, we come to cyclotomic schemes introduced by P. Delsarte (1973), see [1, p. 66]. One can see that any two such schemes of the same valency are isomorphic. Moreover, the automorphism group of such a nontrivial scheme is a subgroup of the group AL(1,F)(see [1, p. 389]). However, there exist a number of cyclotomic schemes over near-fields which are not isomorphic to cyclotomic schemes over fields. The main purpose of this paper is to study isomorphisms of cyclotomic schemes over near-fields.

The additive group of a finite near-fieldKbeing an elementary Abelian one can be identified with the additive group of a linear spaceV_Kover the prime field con- tained in the center of K. The existence of an isomorphism between a cyclotomic scheme over a near-fieldKand a cyclotomic scheme over a near-fieldK, implies that|K| = |K|and hence that the linear spacesV_KandV_K are isomorphic. Thus to study isomorphisms of cyclotomic schemes, we can restrict ourselves to near-fields Kwith a fixed linear spaceV =V_K.

Theorem 1.1 LetCandCbe nontrivial cyclotomic schemes over near-fieldsKand K, respectively. Suppose thatV =V_K=V_K. Then Iso(C,C)⊂AGL(V ). In partic- ular, Aut(C)≤AGL(V ).

For a trivial scheme C, we obviously have Aut(C)=Sym(K). Thus the inclu- sion Aut(C)≤AGL(V )holds only if|K| ≤4. In general, the right-hand side of the first inclusion of Theorem1.1cannot be refined, because Iso(C,C)=AGL(V )for a

schemeC=Cyc(K,F),whereF is a finite field of composite order, andK is the multiplicative group of the prime subfield ofF.

We prove Theorem1.1in Sect.3. The key ingredient for the proof is Theorem3.2 showing that the operation of taking the 2-closure preserves the socle of any uniprim- itive 3/2-transitive permutation groups of affine type. (Here we essentially use the result of [9].) From Theorem1.1we deduce a criterion for the isomorphism of cyclo- tomic schemes (Theorem3.4).

The second part of Theorem1.1can be made much more precise in some cases.

For instance, if the cyclotomic schemeC=Cyc(K,K)is imprimitive, then Aut(C)= Γ (K,K)(Corollary3.5). In general, this equality does not hold even for a cyclotomic scheme over a finite field, because the group Aut(C)can contain some automorphisms of this field. However, we are able to specify the automorphisms of a cyclotomic scheme by using Zsigmondy prime divisors of its valency.

Definition 1.2 Given integersq, n∈N,a prime divisorrofqⁿ−1 is called a Zsig- mondy prime for(q, n)ifr does not divideqⁱ−1 for all 1≤i < n. The set of all such primes greater than a fixed numberk∈Nis denoted byZ_k(q, n).

It is known that at least one Zsigmondy prime for (q, n)exists unless (q, n)= (2,6), orq+1 is a power of 2 andn=2 (see, e.g., [11]). Moreover, any such prime is of the formr=an+1 for somea≥1.

Theorem 1.3 LetC∈Cyc(p^d, n, m)be a cyclotomic scheme over a Dickson near- field andk=dn. Then Aut(C)≤AL(1, p^k)whenevermhas a prime divisorr∈ Z2k+1(p, k).

From Lemma4.2it follows that for a fixedp^d the setZ_2k+1(p, k) is not empty for all sufficiently largek. This fact enables us to prove that the hypothesis of Theo- rem1.3is satisfied in many cases. More precisely, the following statement holds.

Theorem 1.4 Let C ∈ Cyc(p^d, n, m) be a cyclotomic scheme over a Dickson near-field and q = p^d. Then Aut(C)≤ AL(1, qⁿ) for all n q such that

|Z_2dn+1(p, dn)| =1.

Theorem1.4is proved in Sect.4by means of the classification of linear groups with orders having certain large prime divisors given in [4]. We believe that a more delicate analysis of this classification could improve our result to show that given a prime powerqfor all but finitely many Dickson pairs(q, n), the inclusion Aut(C)≤ AL(1, qⁿ)holds for all nontrivial cyclotomic schemesCover a Dickson near-field corresponding to(q, n).

2 Permutation groups and association schemes

2.1

Concerning basic facts of finite permutation group theory, we refer to [3]. LetV be a finite set,Γ ≤Sym(V ),andm∈N. Denote by Orbm(Γ )the set of all orbits of the

induced action of Γ on the setV^m; these orbits are called them-orbits of Γ. The largest subgroup of Sym(V )them-orbits of which coincide with those ofΓ is called them-closure ofΓ; we denote it byΓ^(m).

LetU be a set with at least two elements, and letm≥2 be an integer. Following [9], we say that a permutation groupG≤Sym(V )preserves a product decomposition U^mofV if the latter can be identified with the Cartesian productU^min such a way thatGis a subgroup of the wreath product Sym(U )Sym(m)in product action. Any elementgof the latter group induces uniquely determined permutationsg1, . . . , g_m∈ Sym(U )andσ∈Sym(m)such that

(u1, . . . , um)^g= u^g_iⁱ¹

1 , . . . , u^g_i^im

m

, whereij=j^σ−1. (5) IfGprojects onto a transitive subgroup of Sym(m), then the subgroup of indexmin Gstabilizing the first entry of points ofU^minduces a subgroup of Sym(U )by per- muting the first entries of points ofV =U^m; this subgroup is called the group induced byGonU. The following statement being a special case of result [9, Lemma 4.1]

will be used in Sect. 3. Below a primitive group is called uniprimitive if it is not 2-transitive, and it is called of affine type if its socle is Abelian.

Theorem 2.1 LetG≤Sym(V )be a uniprimitive group of affine type. Suppose that soc(G)=soc(G⁽²⁾). ThenG andG⁽²⁾ preserve a product decompositionV =U^m such that|U| ≥5,m≥2,and the group induced byG⁽²⁾onUcontains Alt(U ).

2.2

LetV be a finite set andRa partition of the setV²containing its diagonalΔ(V )and closed with respect to the permutation of coordinates. The pairC=(V ,R)is called an association scheme or a scheme onV if, given binary relationsR, S, T ∈R,the number

v∈V :(u, v)∈R, (v, w)∈S

does not depend on the choice of(u, w)∈T. The elements ofRand the number|R|

are called the basis relations and the rank ofC respectively. The scheme is called imprimitive if a union of some of its basis relations is an equivalence relation onV other thanΔ(V )andV²; otherwise the scheme is called primitive whenever|V|>1.

Two schemesC=(V ,R)andC=(V,R)are called isomorphic if there exists a bijectionf :V →V,called the isomorphism fromC toC, such thatR^f =R, whereR^f = {R^f : R∈R}withR^f = {(u^f, v^f): (u, v)∈R}. The set of all such isomorphisms is denoted by Iso(C,C). The group Iso(C)=Iso(C,C)contains the normal subgroup

Aut(C)=

g∈Sym(V ): R^g=R, R∈R called the automorphism group of the schemeC.

A wide class of schemes comes from permutation groups as follows. Let Γ ≤ Sym(V )be a permutation group andR=Orb2(Γ ). Then the pair Inv(Γ )=(V ,R) is a scheme and

Aut Inv(Γ )

=Γ⁽²⁾.

In particular, any cyclotomic scheme Cyc(K,K) over a near-field K equals the scheme Inv(Γ )withΓ =Γ (K,K)(see (4)). One can prove that this scheme is prim- itive iff so is the groupΓ.

2.3

Let K be a near-field andK≤K^×. Then the group Γ (K,K)defined by (4) can be naturally identified with a subgroup of the group AGL(V ),whereV =V_K(see Sect.1). Under this identification, the groupK(considered as a subgroup of the group Γ (K,K)) goes to a subgroup of the group GL(V ). This subgroup is called the base group of the cyclotomic scheme Cyc(K,K).

Theorem 2.2 LetCbe a cyclotomic scheme over a near-fieldK. ThenCis primitive iff the base group ofCis irreducible.

Proof LetC=Cyc(K,K)for some groupK≤K^×. Then the schemeCis primitive iff the groupΓ =Γ (K,K)is primitive (see Subsect.2.2). However, from [3, Theo- rem 4.7.A] it follows that the latter statement holds iff the stabilizer of the point 0_K in the groupΓ is an irreducible subgroup of the group GL(V_K). Since this stabilizer coincides with the base group of the schemeC, we are done.

It should be noted that the base group of a primitive cyclotomic scheme Cyc(K,K) can be primitive (as a linear group) or not. For example, it is always primitive for K=K^×, and it is imprimitive forK= {1}if the number|K|is a composite one.

Corollary 2.3 The cyclotomic schemeCin Theorem1.3is primitive.

Proof LetGbe the base group of the schemeC. ThenGis a solvable subgroup of the group GL(k, p), andrdivides the ordermofG. By [6, Proposition 6.3] this implies that the groupGis irreducible. Thus the schemeCis primitive by Theorem2.2.

LetV be a finite dimensional linear space over a finite prime field, and letG≤ GL(V )be an irreducible Abelian group. ThenGis a cyclic group and its linear span L(G)in the algebra End(V )is a finite field with|V|elements (see [6, Lemma 0.5]).

The multiplicative group of this field acts regularly on nonzero vectors ofV, i.e., this group is a Singer subgroup of the group GL(V ). So, given a fixed nonzerou₀∈V , the mapping

τ:L(G)→V , A→Au₀,

is a bijection. This defines a field F=F(G) with elements from V such that F⁺ coincides with the additive group of the linear spaceV. Clearly,τ (G)≤F^×. Theorem 2.4 Any primitive cyclotomic scheme with Abelian base group is a cyclo- tomic scheme over a field.

Proof LetC=Cyc(K,K)be a primitive cyclotomic scheme andV =V_K. Suppose that its base groupG≤GL(V )is Abelian. Then Gis irreducible by Theorem2.2.

This enables us to construct the fieldF=F(G). From the definition of this field it follows thatF⁺=K⁺and

x◦y=x·y, x∈V , y∈M,

whereM=τ (G), and◦ and·denote the multiplications inKandF,respectively.

This implies thatΓ (K,K)=Γ (M,F),and henceC=Cyc(M,F)is a cyclotomic

scheme over the fieldF.

3 An isomorphism criterion for cyclotomic schemes

3.1

In this section, we prove Theorem1.1. For cyclotomic schemes with primitive base group we will use Theorem2.1. In the imprimitive case, we need an auxiliary result on 3/2-transitive groups, where by such a group we mean a transitive permutation groupΓ for which the orbits of its one point stabilizerΓvother than{v}all have the same size.

Lemma 3.1 LetG≤Sym(V ) be a 3/2-transitive group preserving a product de- compositionV =U^mform≥2. Then the stabilizerGu,vof some pointsu, v∈V is an Abelian 2-group.

Proof Letu∈V andI= {1, . . . , m}. Without loss of generality we may assume that u=(u0, . . . , u0)∈U^mfor someu0∈U. Then from (5) it follows thatu^g₀ⁱ =u0for allg∈Guand alli∈I. So the cardinality of the setIv= {i∈I: vi=u0},wherevi

is theith component ofv∈V, does not depend on the choice ofvinside of an orbit of the groupG_u. Thus, the sets

Vk=

v∈V : |Iv| =k

, k=1,2, (6)

R=

(v, w)∈V1×V2: v_i=w_i for the uniquei∈I_v

areG_u-invariant. Obviously,|R_in(w)| =2 for allw∈V₂whereR_in(w)= {v∈V : (v, w)∈R}. We divide the remaining argument into a sequence of claims.

Claim 1 LetX∈Orb(Gu, V₁),Y∈Orb(Gu, V₂),andS=R∩(X×Y ). Then Sout(x)≤2, x∈X,

whereSout(x)= {v∈V : (x, v)∈S}. Indeed, sinceSis aG_u-invariant relation, the numbers|S_out(x)|and|S_in(y)| do not depend onx∈X andy ∈Y, respectively. If we denote them bya andb, then obviously |X|a= |Y|b. Taking into account that

|X| = |Y|due to 3/2-transitivity ofG, we conclude thata=b≤2 (see the above remark).

Claim 2 Letxandy be elements ofV₁such thatI_x=I_y. Then y^Gu,x≤2.

Indeed, let I_x = {i} and I_y= {j} for some distinct i, j ∈I. Then there exists a uniquely determined element w∈V₂ such thatx_i =w_i andy_j =w_j. Denote by XandY the orbits of the groupG_ucontainingxandw,respectively. From Claim1, it follows thatSout(x)= {w, w}for somew∈Y. Since the setSout(x)is obviously G_u,x-invariant, we conclude that so is the setRin(w)∪Rin(w). However, this set contains at most three elements two of which arexandy. Thus

y^Gu,x≤R_in(w)∪R_in(w)

\ {x}≤2, which proves the claim.

Claim 3 Let(x, w)∈V1×V2. Then the transitive constituentH of the groupGu,x

induced by its action on the set Y =w^Gu,x is a 2-group. Indeed, without loss of generality we may assume that |Y|>2 andI_w= {i, j}for some distincti, j ∈I. Theni, j∈I_x, since otherwiseY ⊂R_out(x)and hence|Y| ≤2 by Claim1. Sety to be the unique element ofV₁\ {x}such thaty_i=w_i. By Claim2the setX=y^Gu,x consists of (not necessary distinct) elementsy, z∈V1, whence by Claim1it follows that

Y =Sout(y)∪Sout(z), 1≤Sout(y)=Sout(z)≤2.

Since|Y|>2 and|Sin(w)| = |Sin(w)|for allw∈Y, we conclude thatSout(y)and Sout(z)are disjoint blocks of the groupH, and each of them is of size 2. This implies thatHis a 2-group isomorphic to a subgroup of the group Sym(2)Sym(2).

Claim 4 The action ofG_u onV₂is faithful. Indeed, anyg∈G_u is of the form (5).

Suppose thatw^g=w for allw∈V2. Then, giveni∈I and all w∈V2 such that {i, j} ⊂I_wandw_i=w_jwherej=i^σ, we have

w_j=(w_ij)^gij =(w_i)^gi=(w_j)^gi.

This implies thatg_i=idU for alli∈I. Next, ifσ=idI, then obviouslyw^g=wfor allw∈V2such thatI_w= {i, j}andw_i=w_j,wherej =i^σ. Thusg=idV,and we are done.

To complete the proof of Lemma3.1takev∈V1. Denote byKthe direct product of transitive constituents of the groupG_u,v corresponding to its orbits contained in the setV₂. ThenKis a 2-group by Claim3. On the other hand, by Claim4the group G_u,vis isomorphic to a subgroup of the groupK. ThusG_u,vis a 2-group.

Theorem 3.2 LetG≤Sym(V )be a uniprimitive 3/2-transitive group of affine type.

Then soc(G)=soc(Γ ),whereΓ =G⁽²⁾.

Proof Suppose that soc(G)=soc(Γ ). Then from Theorem2.1it follows that the groups G and Γ preserve a product decomposition V =U^m such that |U| ≥5,

m≥2,and the group induced byΓ onU contains Alt(U ). This implies that

|Γ| =amAlt(U ) (7)

for somea∈N. On the other hand, the groupΓ obviously is 3/2-transitive. Denote bydthe size of an orbit of its one point stabilizerΓ_vother than{v}. Then it is easy to see thatd=mefor some divisoreof|U| −1 (it suffices to check the orbit of a point from the setV₁defined in (6)). By Lemma3.1forG=Γ this implies that

|Γ| = |V|me2^k (8) for somek∈N. Thus equalities (7) and (8) show that|Alt(U )|divides|V|e2^k. Since edivides|U| −1, it follows that(|U| −2)!divides|V|2^k+1. However, this is impos- sible for|U| ≥5, since|V|is a prime power (we used the fact thatGis of affine

type).

From Theorem3.2it follows thatG⁽²⁾ is a uniprimitive 3/2-transitive group of affine type. If in addition, the groupGpreserves a product decomposition, then the same decomposition is preserved byG⁽²⁾. Thus, in this case, the form of this group can be found by means of the classification of 3/2-transitive imprimitive linear groups given in [8].

3.2

In this subsection we fix a near-fieldKand a cyclotomic schemeCoverKand denote byT =TV the translation group of the linear spaceV =V_K. Clearly,T ≤Sym(V ).

Lemma 3.3 If the schemeCis nontrivial, thenT is a characteristic subgroup of the group Aut(C). More exactly, the following statements hold:

(1) IfCis imprimitive, then Aut(C)is a Frobenius group with kernelT. (2) IfCis primitive, thenT =soc(Aut(C)).

Proof LetC=Cyc(K,K)andΓ =Γ (K,K),whereK <K^×(see (4)). ThenC= Inv(Γ )and so Aut(C)=Γ⁽²⁾. On the other hand, it is easy to see that the orbits of the groupΓ_vother than{v}all have the same size|K|. This implies that the groupΓ and hence the group Aut(C)is 3/2-transitive.

Let C be an imprimitive scheme. Then the group Aut(C)is imprimitive. Since any 3/2-transitive group is either primitive or a Frobenius group [14, Theorem 10.4], it follows that Aut(C) is a Frobenius group. The kernel of this group is of order

|V| = |T|and contains all fixed-point-free elements of the groupΓ. Thus the kernel coincides withT ,which proves statement (1).

LetC be a primitive scheme. Then the group Γ is primitive andT is a normal Abelian subgroup of it. This implies that the socle ofΓ is Abelian and hence coin- cides withT (see [3, Theorem 4.3.B]). ThusΓ is a uniprimitive 3/2-transitive group of affine type. By Theorem3.2this implies that

T =soc(Γ )=soc Γ⁽²⁾

=soc Aut(C)

,

which completes the proof of the lemma.

Proof of Theorem 1.1 Letf ∈Iso(C,C). Then the bijectionf induces an isomor- phism between permutation groups Aut(C)and Aut(C). Since these groups are tran- sitive, without loss of generality we may assume thatf leaves the point 0∈V fixed.

Then it suffices to verify that f belongs to the group Aut(T )=GL(V ). However, the schemesCandCand hence the groups Aut(C)and Aut(C)are primitive or not simultaneously. Thus the required statement follows from Lemma3.3.

3.3

To make the statements of Theorem1.1more precise, given a groupG≤GL(V ),we set

G=G⁽¹⁾∩GL(V ). (9)

Clearly,Gcoincides with the largest groupH≤GL(V )such that Orb(H )=Orb(G).

Theorem 3.4 Under the conditions of Theorem1.1, denote byGand G the base groups of the schemesCandC,respectively. Then these schemes are isomorphic iff the groupsGandGare conjugate in GL(V ). Moreover, Aut(C)=T G.

Proof The first part of the theorem follows from the second one. Indeed, setΓ = Aut(C)andΓ=Aut(C). Then by Theorem1.1the schemesCandCare isomorphic iff there existsg∈GL(V )such thatg⁻¹Γ g=Γor, equivalently, thatg⁻¹Γvg=Γ_v wherevis the zero vector of the linear spaceV. Since by the second partΓv=Gand Γ_v=G, we are done.

To prove the second part of the theorem we note that from Theorem1.1it fol- lows thatΓ =T Γ_vandΓ_v≤GL(V ). Since obviously Orb(Γv)=Orb(G), we con- clude thatΓ_v≤G⁽¹⁾and Orb(Γv)=Orb(G). This shows that Orb2(Γ )=Orb2(T G), whence by maximality of the 2-closure it follows thatT G≤Γ. Thus Aut(C)=Γ =

T G,and we are done.

For imprimitive cyclotomic schemes, Theorem3.4can be slightly simplified. In- deed, in this case, Aut(C)is a Frobenius group by statement (1) of Lemma3.3. So

|G| =Aut(C)_v= |X| = |G|,

whereXis an orbit of the group Aut(C)_vother than{v}. Since alsoG≤G, we have G=G. Thus by Theorem3.4we obtain the following statement.

Corollary 3.5 Let the cyclotomic schemes C andC be imprimitive. Then they are isomorphic iff their base groups are conjugate in GL(V ). Moreover,G=G and Aut(C)=T G.

4 Proof of Theorems1.3and1.4

The main tool of this section is Theorem4.1below which is deduced from the clas- sification of linear groups with orders having certain large prime divisors [4]. In our

case such a divisor is a Zsigmondy prime r for a pair (q, n), whereq is a prime power andn∈N. Any cyclic groupG≤GL(n, q)of orderris irreducible [6, Propo- sition 6.3], and the linear span L(G) of it in Mat(n, q)is a finite field F withqⁿ elements. We will identify the groupL(1,F)with a subgroup of GL(n, q). Below a group Γ ≤GL(n, q) is called half-transitive if the action of it on the setV^∗ of nonzero vectors in the underlying linear space is intransitive and the orbits of this action all have the same size.

Theorem 4.1 LetG≤Γ ≤GL(n, q)where(q, n)∈ {(2,4), (2,6)}. Suppose thatG is a cyclic group of order r∈Z_2n+1(q, n)and that the groupΓ is half-transitive.

ThenΓ ≤L(1,F),whereF=L(G).

Proof It suffices to prove that Γ ≤L(1, qⁿ). Indeed, in this case Γ ≤L(1,F) for some fieldF⊂Mat(n, q)withqⁿ elements. So the multiplicative group ofF normalizesGand hence normalizes the Singer subgroupF^×⊂L(G) of the group GL(n, q). However, the normalizer of F^× in GL(n, q)contains the unique Singer subgroup [2, Proposition 2.5]. This proves thatF=F.

Suppose thatΓ is a solvable group. Ifrdivides the order of the Fitting subgroup of Γ, then this group is isomorphic to a subgroup of L(1, qⁿ) [6, Lemma 6.4].

Otherwise from Lemma 6.7 of the same book it follows thatr=n+1,which con- tradicts the hypothesis onr. Thus the required statement is true for solvable groups.

In particular, we may assume thatn≥2 and that the groupΓ is nonsolvable.

Letn=2. From the classification of all subgroups of GL(2, q)given in [7, Propo- sition 8.1] it follows that any nonsolvable irreducible subgroup of GL(2, q), sayΓ, has a subgroupHsuch that[Γ :H]dividesq−1 and

H≥SL(2, q) or H /Z∼=Alt(5),

whereq≥5 is a divisor ofq, andZ is the subgroup of scalar matrices contained inH. However, in the former case, this group acts transitively on the setV^∗, and the intransitivity of Γ gives a contradiction. In the latter case, the prime divisors of the number|Γ|are over those of the number(q−1)5!/2, which contradicts the assumption thatr∈Z_2n+1(q, n)forn=2. It should be mentioned that in this case we proved the required statement for the groupΓ which is intransitive but not necessary half-transitive.

Letn≥3. The Zsigmondy primerfor the pair(q, n)is a primitive prime divisor ofqⁿ−1 in the sense of [4]. Sincer divides the order of the groupΓ, this group satisfies the hypothesis of the Main Theorem of that paper ford=e=n. In this case, the Main Theorem shows that, for the group Γ (not necessary half-transitive) and r >2n+1,one of the following statements holds:

(1) Γ has a normal subgroupΓisomorphic to one of the classical groups SL(n, q), Sp(n, q), SU(n, q^1/2),orΩ (n, q),whererdivides the order ofΓ,qis the order of a subfield of the ground field, and ∈ {◦,+,−},

(2) Γ ≤ GL(n/m, q^m)·m, and the number r divides the order of the group Γ ∩GL(n/m, q^m),wheremis a divisor ofnother than 1,

(3) (q, n)=(2,4)or(2,6),

where GL(n/m, q^m)·m is the general linear group GL(n/m, q^m) embedded to GL(n, q) and extended by the group of automorphisms of the field extension GF(q^m)/GF(q). However, the case (3) does not arise by the hypothesis of the theo- rem. Let us prove that the same is true in the other two cases.

We claim that the groupΓ contains a normal nonsolvable subgroupH0isomorphic to one of the groups SL(n0, q0), Sp(n0, q0), SU(n0, q₀^1/2),or Ω (n0, q0),wherer divides the order ofH0,n0≥2 is a divisor ofn, andq0is the order of a subfield of the field GF(q^n/n0). Indeed, in case (1) we can takeH₀=Γand(n₀, q₀)=(n, q).

Otherwise, case (2) holds. It is easy to see thatΓ₀=Γ ∩GL(n/m, q^m)is a normal subgroup ofΓ andΓ /Γ₀is a cyclic group of order coprime tor. This implies that the groupΓ has a characteristic subgroupH≤Γ0such that the factor groupΓ /His solvable and each prime divisor of its order dividesm. In particular,

G≤H≤GL

n/m, q^m ,

andH is solvable iffΓ is so. If case (2) holds for the groupH, we repeat this ar- gument withΓ =Hand(n, q)=(n/m, q^m). Finally, we find a nonsolvable charac- teristic subgroup H of the group Γ such thatG≤H≤GL(n1, q1),wheren1≥2 is a divisor ofn,andq₁is the order of a subfield of the field GF(q^n/n1). Moreover, we may assume that case (1) holds forΓ =H and(n, q)=(n₁, q₁). Since the cor- responding classical groupΓ is a characteristic subgroup ofH, we are done with H0=Γand(n0, q0)=(n1, q₁).

To complete the proof we will show that the above claim contradicts the half- transitivity ofΓ. Without loss of generality we assume thatn0≥3 (see above). Then the groups SL(n0, q₀)and Sp(n0, q₀)act transitively on the setV^∗[5, Lemma 2.10.5].

By the intransitivity ofΓ this implies thatH₀cannot be one of these groups. There- fore we may assume thatH₀is either the unitary group SU(n0, q₀^1/2)or the orthogonal groupΩ (n₀, q₀). Given an elementλof the ground fieldF, set

V_λ^∗= v∈V^∗: f (v, v)=λ

ifH0=SU

n0, q₀^1/2 ,

v∈V^∗: Q(v)=λ

ifH0=Ω (n0, q0),

wheref (resp.Q) is the nondegenerate unitary (resp. quadratic) form corresponding toH0. By the same lemma, forn0≥3,we obtain that

Orb H0, V^∗

=

V_λ^∗: λ∈F

unlessn0=3, H0=Ω(3, q0), λ=0, and in the exceptional case the set V_λ^∗ is the union of two H0-orbits each of size (q₀²−1)/2. Clearly, the numbera_λ= |V_λ^∗|does not depend onλ=0. So using the explicit formulas fora₀(see Lemma 10.4 and Theorem 11.5 in [12]), one can see that q0is coprime toa0and dividesa_λfor allλ=0. SinceΓ acts on the set Orb(H0, V^∗) (due to the normality ofH0inΓ), this implies that

V₀^∗Γ

=V₀^∗ and Orb

Γ , V₀^∗≤2.

By the half-transitivity ofΓ this shows that the size of anyX∈Orb(Γ , V^∗)is co- prime toq0. However, this contradicts the fact thatq0divides|X|for allX⊂V₀^∗, and

we are done.

Proof of Theorem1.3 The hypothesis shows thatr divides the ordermof the base group of the schemeC. So this group contains a cyclic subgroupGof orderr. By Theorem1.1we have

G≤Γ ≤GL(k, p),

whereΓ =Aut(C)_vwithv=0. Besides, the orbits in the action ofΓ onV^∗all have the same sizem < p^k−1, and hence the groupΓ is half-transitive. Thus by Theo- rem4.1with(q, n)=(p, k)it suffices to check only the cases(p, k)∈ {(2,4), (2,6)}.

However, the fact that(2^d, n)withdn∈ {4,6}is a Dickson pair implies that either (d, n)=(2,3)orn=1. But, in the former casek=6 andZ_2k+1(2, k)= ∅, whereas in the latter case the near-field is a field, and we are done (see Sect.1).

The following auxiliary lemma is a combination of some number theoretical re- sults from [10] and [11]; it will be used in the proof of Theorem1.4.

Lemma 4.2 Given a prime powerq=p^d,there exists an integerNq∈Nsuch that the setZ2dn+1(p, dn)is not empty for alln > Nq.

Proof Givenn∈N, denote byD(n)the number of distinct prime factors ofn, by P[n]the greatest of them, and byΦn(X)the cyclotomic polynomial of degreeϕ(n).

Then there exists a constantC >0 such that P

Φ_dn(p) > Cn

logn/log log logn (10)

for all sufficiently largensuch thatD(dn)≤κlog log(dn)withκ=1/(2 log 2)(see [10, p. 25]).

Denote byS_qthe set of all integersdn∈N such that(q, n)is a Dickson pair and n is greater than the minimal numbera∈Nfor whichD(d)logq ≤κlog log(da).

Then givendn∈S_q, we have

D(dn)≤D(d)D(n)≤D(d)logq≤κlog log(dn).

By (10) this implies thatP[Φ_dn(p)]>2dn+1 for all sufficiently largen∈S_q. How- ever, a prime factor rof the numberΦ_dn(p)is not a Zsigmondy prime for (p, dn) iffr≤dn(see [11, Proposition 2]). Thus there exists a positive integerN_q such that

Z2dn+1(p, dn)= ∅for alln > N_q.

Proof of Theorem1.4 Let C=Cyc(K,K^×),whereKis a Dickson near-field cor- responding to the Dickson pair(q, n),and the groupK≤K^×is of orderm < qⁿ. Suppose thatn > Nq. Then by Lemma4.2the setZ2dn+1(p, dn)is not empty. Set Kto be a maximal subgroup ofK^×containingK. Since the groupK^×≤L(1, qⁿ) is supersolvable, the number [K^×:K] is prime. So if|Z2dn+1(p, dn)| =1, then the numberm= |K|has a prime divisorr∈Z2dn+1(p, dn). This implies that the scheme C=Cyc(K,K^×)belongs to the class Cyc(q, n, m)and satisfies the hy- pothesis of Theorem1.3withrreplaced byr. Thus

Aut(C)≤Aut(C)≤AL 1, qⁿ

, (11)

and we are done.

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