A N T O N I N O G I O R G I O S P E R A

S E M I - R E G U L A R D I V I S I B L E D E S I G N S A N D F R O B E N I U S

G R O U P S

ABSTRACT. We study and characterize semi-regular (s, k, 3,1, )~2)-divisible designs which admit a Frobenius group as their translation group. Moreover, we give a construction method for such designs by generalized admissible triads.

1. I N T R O D U C T I O N

Divisible designs with parameters s, k, ).1 and ).2 (or briefly (s, k, ).1, )-2) -DD's) admitting a 'nice' automorphism group have been considered in several papers. In some of them (see, for instance, [2], [4], [6], [8] and [9]) there were studied (s, k, ),1, ).2) -DD's admitting a Frobenius group as a translation group in the case ).1 = 0. It was shown (see [8]) that such a structure must be transversal, and examples were constructed whether the kernel of the Frobenius group is abelian or not. In the present paper, we are interested in studying semi-regular (s, k, 21, 22) -DD's admitting a Frobenius group as a translation group in the case )-1 ~ 0. Following [2], [8] and [9], we make use of group partitions and generalized admissible triads to describe them. In Section 2 we give some preliminary results about the structure of the translation group. In Section 3 we characterize such structures by means of generalized admissible triads and in Section 4 we give a construction of semiregular (s, k, 21, )-2)-DD's whose translation group is a Frobenius group with abelian kernel.

2. D E F I N I T I O N AND F I R S T P R O P E R T I E S

A divisible design with parameters s, k, 21 and )-2 (or an (s, k, 2t, ).2) -DD) is a finite incidence structure D = (g~, ~) satisfying the following conditions.

(1) The point set g~ is split into disjoint classes of size s such that any two distinct points in the same point class are incident with exactly ).1 blocks, whereas any two points in distinct classes are incident with

exactly ).2 blocks ().1 and )-2 both not zero and s < 1#1). (2) Every block is incident with exactly k points (1~1 > k > 1).

If v = I~1 and b = I~1, we have the following

P R OP OS ITI ON 2.1 (Bose and Connor [3]). Let D be an (s,k, 21,22)-DD.

Geometriae Dedicata 42: 285-294, 1992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands.

286 ANTONINO GIORGIO SPERA

Then each point is incident with exactly r blocks and

(i) r ( k - 1) = ( v - s ) 2 2 + ( s - 1)2~ (ii) rv = bk and rk >~ VI~ 2.

In the case rk = v22 and r > 21, the divisible design is called semi-regular and

we have

P R O P O S I T I O N 2.2. The following properties of an (s, k, 21, 22)-DD D are equivalent:

(i) D is semi-regular; (ii) v divides sk and every block meets every point class in exactly sk/v

points;

(iii) r = s ( 2 2 - 2 1 ) + 2 1 and 22 > 21.

Proof. We can suppose, in the semi-regular case, that s > 1 since k < v. We can also suppose 21 # 0, because for 21.= 0 the propos i t ion can be found in

[10]. Let E be a fixed point class and consider the incidence structure A = (E, ~ ' ) where ~ ' is the indexed set {B i c~ E IBi ~ ~} . If D verifies (ii), then A is a block design with pa ramete r s v'= s, k '= sk/v, r ' = r and 2 ' = 21. (Note that A m a y have repeated blocks.) So ( s - 1)21 = r(sk /v- 1) and, by (i) of Propos i - t ion 2.1, it follows by an easy calculat ion that rk=v2 v Thus (ii) implies (i). The converse is Theo rem 4 in 1-3] (see also 1-7, Prop. 1.4]); thus we have that (i) is equivalent to (ii). Moreover , by (i) of Propos i t ion 2.1, we also obtain that (i) is equivalent to (iii) after some calculation.

Fo r an (s, k, 21, 22)-DD we will always suppose that s > 1 and

0 # 2 1 # 2 2 # 0 .

If a divisible design D admits an a u t o m o r p h i s m group G which acts regularly on the point set in such a way tha t B#c~ B = ~ or B g = B for every g ¢ G and B z ~ , then we call D a translation divisible design (TDD) and G a t ransla t ion group of D. Later on, for T D D ' s we do not admit repeated blocks.

Now, let G be a finite group, H a non-tr ivial subgroup of order s of G and R = {G1, G2 . . . . , G,} a set of non-tr ivial subgroups of G, all of the same order k. R is called an H-(s, k, 21, 22)-partition of G, where :.1 and 22 are distinct positive integers, if the following condi t ion holds:

Every element x of G - { 1} is contained in exactly 21 components of

R if x ~ H, and in exactly 22 components of R if x ~ G-- H.

For an H-(s, k, 2 , 22)-partition we will always assume tha t its componen t s are distinct unless the con t ra ry is stated. We omit the p roof of the following

P R O P O S I T I O N 2.3. I f R is an H-(s, k, 21, 22)-partition of a finite group G and

S E M I - R E G U L A R D I V I S I B L E D E S I G N S 287

D(R) denotes the incidence structure whose point set is G and block set is {Gix [ x • G and Gi• R}, then D(R) is an (s, k, ~,1, 22)-TDD with translation group isomorphic to G. Conversely, if D' is an (s, k, 21, Z2)-TDD with translation group G', p a fixed point of D' and B1, B E , . . . , B r the blocks through p, then R ' = {G~, G~ . . . . , G',} is an H'-(s, k, ~1, 22)-partition of G' and D(R') ~- D', where H' is the stabilizer of the point class containing p and G'i is the stabilizer of B~ for all i= 1,2 . . . . , r.

Often we will call the T D D D(R) defined above by an H-(s, k, ),1, ,~2)-P art i t ion R of G, the associated design of R and we will also call R semi-regular if D(R) is semi-regular. We recall that a non-trivial par t i t ion of G is a set of non-tr ivial

subgroups of G which cover G and such that the intersection of any two distinct componen t s is trivial.

P R O P O S I T I O N 2.4. I f R = {G1, G 2 , . . . , Gr} is a semi-regular H-(s, k, 21, 22)- partition of G and we set Go = H, then for every i = O, 1 , . . . , r, G~ admits a non- trivial partition and is a component of a non-trivial partition of G.

Proof. Let Lx, x • G - { 1 } , be the intersection of all the componen t s of R

containing x. Of course Lx is a non-trivial subgroup of G. If x • Go, Lx c~ Go is also a p roper subgroup of G o because in the cont rary case Go - Lx _c G~ for some 1 ~<i~< r and so, by (ii) of the Propos i t ion 2.2, G o_c G~ for every i = 1, 2 , . . . , r. Whence it follows that r = 21 contradict ing the semi-regularity.

Since, if x, y • Go, (L~ n Go) c~ (Lr n Go) ~ { 1} implies L~ n Go = Ly c~ Go, we obtain that rt(H) = {L~ c~ Go [ 1 ~ x • Go} is a non-trivial par t i t ion of G 0. Now, suppose x ~ Go. There are/~2 componen t s of R containing x. So, if y • L~ n Go, y is contained in ~2 components . But, by (iii) of Propos i t ion 2.2,/~2 ~ )~!" Thus y = 1 and we have that L~c~Go = {1}. Therefore

~(a i ) = { a 0 ~ ai} u {L x I x • a i - Go}

is a non-trivial par t i t ion of G~, since for x , y • G i - G o and L~,c~Ly ~ {1} we

have L x - - L r A similar a rgument for

= 0 r ,

gives tha t n~ is a non-trivial part i t ion of G containing G~, and so the propos i t ion is shown.

By a Frobenius group with complement C we unders tand a finite g roup G with a non-tr ivial subgroup C such that C c~ x-1 Cx = { 1} for every x • G - C . It is well known that the Frobenius kernel

K = { g e G l g e x - X C x for every x • G } u { 1 }

of G is a normal subgroup of G which is a p-group (see [-11]) in the case where

288 A N T O N I N O G I O R G I O SPERA

G has a non-tr ivial par t i t ion different f rom the following Frobenius par t i t ion

(1) rc F = {K} w { x - l C x l x e K } .

T H E O R E M 2.5.Let G be a Frobenius #roup. I f R = {Gi, G2, . . . , G,} is a semi- regular H-(s, k, 21, 22)-partition of G, then H is contained in the Frobenius kernel K of G and, for every i = 1, 2 , . . . , r, Gi is a Frobenius #roup with complement C x where C is a complement of G and x is a suitable element of K.

Proof. By Propos i t ion 2.4, G admits some non-trivial par t i t ions different f rom the Frobenius part i t ion, so the Frobenius kernel K of G is a p-group. I t is well known (see [1]) that C x = x - 1 C x admits only the trivial part i t ion, so if C x n Gi # { 1 }, then C x ___ Gi because Gi is a c o m p o n e n t of some par t i t ion of G. The same for H, that is H c~ C ~ ~ { 1 } implies C ~ ___ H. Now, if y # 1 and y e C, then there exists at least one c o m p o n e n t Gi of R containing y. So C ___ G~ and

we have that c = ICI divides k. Of course k ~ c since 22 > 21 > 0. Thus if I K [ = p m, k=pac with d < m. If we set K i= G i n K for i = 1 , 2 , . . . , r , we have that K~ is a p-Sylow subgroup of G~ of order pd and so G~ is a Frobenius group with kernel K~ and complemen t C ~ for some x e K. N o w we suppose tha t

H ~ K. Then, since H is a c o m p o n e n t of some non-tr ivial par t i t ion (see Propos i t ion 2.4) of G, it is a Frobenius group too. In fact, H is not a Frobenius complement because it has (again by Propos i t ion 2.4) a non-tr ivial part i t ion; thus there exists a complemen t C ~ which is a p roper subgroup of H

and such that (CX)YnCX={1} if y ~ H - C ~. So s=lHl=pUc, where p~ = [H n K[ and u > 0. Set K o = H n K and consider the incidence structure D(K) = ( ~ , ~ ) defined by {K 1, K 2 . . . . . Kr} and K o in the following way. (Note that the K~'s are not necessarily distinct, so they do not necessarily form

a Ko-(p u, pa, 2~, 22)-partition of K.) ~ = K, ~ = {Kix I i = 1, 2 . . . . . r and x e K}. I t is easy to show that D(K) is a (pU, pal, 21, 22)-DD where the point classes are the right cosets of K o. So, by (i) of Propos i t ion 2.1, we have

(2) (pm_p,)22 + ( p U 1)21 = r(pn_ 1).

A similar relat ion can be wri t ten down for the (p~c, pdc, 21, 22)-TDD D(R):

(3) (p=c -pUc)22 -}- (pUc- 1)21 = r ( p d c - 1).

Now, combining (2) and (3), we obtain, after rearrangement ,

(22 - 21)- l[r(c - 1 ) - (21(c - 1)] = 0,

a n d s ince 2 2 > 21 we h a v e

(4) r(c-- 1)--21(c-- 1) = 0.

SEMI-REGULAR DIVISIBLE DESIGNS 289

But D(R) is semi-regular and so r > ~,1, that is r - 2 1 ~ 0. Thus f rom (4) we obtain

(5) c - 1 -- 0.

Of course, (5) contradicts the fact that c = ICI where C is a Frobenius complemen t of G. Therefore H must be contained in K and the theorem is shown.

3. FROBENIUS GROUPS AND GENERALIZED A-ADMISSIBLE TRIADS

In this section we will characterize, following [2] and [8], semi-regular H- (s, k, 21, 22)-partitions of Frobenius groups (and so, by Propos i t ion 2.3, semi- regular (s, k, 21, 22)-TDD's with a Frobenius group as t ranslat ion group) by generalized A-admissible triads.

In the case where R is an H-(s, k, 21, 22)-partition of G whose componen t s are not necessarily distinct, we will write R = [G1, G 2 , . . . , G,1 if G1, G 2 , . . . , G r are the componen t s of R. Of course, also in this case, it is possible to associate with R an (s, k, 21, 22)-DD which is defined as in Propos i t ion 2.3. Thus we will again call R semi-regular if such a divisible design is semi- regular.

N o w let K be a finite g roup and ~¢ = [Lo, L 1 , . . . , L,] (r >/ 1) a multiset of

subgroups Lo, L 1 , . . . , L, of K, that is a set of subgroups of K which are not necessarily distinct. Let A be a non-trivial fixed point free (f.p.f.) a u t o m o r p h -

ism group of K and let q ) : d - { L o } ~ L o be a map. If we set ui=L'r for i = 1, 2 , . . . , r, then the triplet (K, ~ , q)) is called a generalized A-admissible triad on K with parameters s, k, 21 and 22 (1 < s, k < [K[ and 2 x, 22 distinct positive integers) if the following condit ions hold:

(1) rui- 1LlU 1, U21Lzu2,..., U~-lLrur] is a semi-regular Lo-(S ,It, 21,,~.2)-

part i t ion (possibly with repeated components) of K.

(2) For every x E K there exist exactly 22 componen t s of

[LlUl, L2u2 . . . . . L,u,] containing x. (3) If u?~ lLiu l = uj-lLjuj and i ¢ j , then Liu i ~ Liuj. (4) Li is invar iant under A for all i -- 1, 2 , . . . , r.

Now, let G be a Frobenius group and let R = {G1, G 2 , . . . , Gr} be a semi- regular H-(s, k, 21, 22)-partition of G. If K is the Frobenius kernel and C a complement , as in the previous section, we set K~--G~c~K for i = 1, 2 , . . . , r and Ko = H n K . By Theorem 2.5, H c_ K and every Gi, i ¢ 0, is a Frobenius group with complement a suitable complement of G. So [Ko] = [H[ = s and IK~[ = pd if k = [G~I = pac and c = [CI. Moreover , for every i = 1, 2 , . . . , r, by (1)

290 A N T O N I N O G I O R G I O S P E R A

we can choose a complement of Gi of the form C"' where u~ ~ K. Thus we have

Gi= K i u ( U C "'~) withu , e K . \ x e K i

It is possible to choose ul, u2 , . . . , u, in Ko. In fact, consider the T D D D(R) associated with R. Since R is semi-regular, D(R) is semi-regular and so, by

Proposi t ion 2.2(ii), IHnG~l=sk/v if v = l G I . But IHnGII=IHIIGil/IHGi I = sk/IHGil. Then IHGzl = v and so G = HGi for any i = 1, 2 . . . . . r. F r o m this we have ul -- hfg~ with h~eH = Ko and gg~ G~. But g~ = hT~u~eK since u~eK. So we have

Gi=G,'=Kiu(U C"'xg')=Ki~(U Ch" ''~'') withhi~K o. \ x E K i / \•EKi /

Therefore we can choose for G~ a complement C"' with ui e Ko obtaining

(6) Gi = K i U ( U cx I with ui~Ko for all i = 1,2 . . . . . r. \ x~u iK i /

N o w set Lo = K0, Li = uiK~ui- 1 for i = 1, 2 . . . . . r and define ~o: [L1 . . . . ,L,q--*L o by L~ = u i where u~ is chosen as above. Of course [u• i L~u~li= 1, 2 . . . . . r] is a Lo-(S, k, ;h, 22)-partition of K with k = k/ICI. Moreover , by the semi-regularity of R, risk. But v = l K I ICl and k = k l C I .

Thus Igl [sk. Moreover ,

]Lonui- ~Liui] = ]H nKi] = ]H nGi[ = sk/v = sk,/lK[.

This is enough to be [u~- ~L~ug] i = 1, 2 . . . . . r] semi-regular, and so (1) of the

definition of triad is shown. (3) follows from the fact that the components of R

are distinct. If x e K, then C ~ ~ K ~ K o. So there exists exactly 22 compo-

nents of R containing C ~. Then, by (6), x is contained in exactly 22

components of [u~K~ = L~u~]i = 1, 2 . . . . , r] and (2) is true. Finally, we remark that C operates f.p.f, on K, and since G~ is a Frobenius group with kernel Ki

and complement C% C ~' normalizes Kz. So C normalizes L~ for all

i = 1, 2 , . . . , r, which yields (4). Thus we have proved the following

P R O P O S I T I O N 3.1. Let G be a Frobenius group with kernel K and complement C. I f R is a semi-regular H-(s, k, 21, 22)-partition of G, then K admits a generalized C-admissible triad (as described above) with parameters s,

= k/ICI, 2x and 2 2.

For the reversion of the above proposi t ion we recall that if A is an

au tomorph i sm group of a group K, the ho lomorph of K by A is the subgroup G(K, A) of the symmetric g roup on K defined by G(K, A ) = {f,,bl a e A and

b e K } where fa,b : K ~ K is defined by (x) f.,b = x"b for every x e K.

S E M I - R E G U L A R D I V I S I B L E D E S I G N S 291

P R O P O S I T I O N 3.2. Let (K, ~ , cp) be a generalized A-admissible triad with parameters s, Ic, ).1 and ).2. Then the holomorph of K by A is a Frobenius group admitting a semi-regular H-(s, k, 21, 22)-partition where k = klAI.

Proof.. Since A is f.p.f, au tomorph i sm group of K, it is easily seen (cf. [2] or [8"1) that G : = G(K,A) is a Frobenius group with kernel

K ' = { f l . b l b e K } " ~ K and complement A, so that G = A K ' . Now, if

~¢ = [Lo, L1 . . . . . L,], we set L'i = {fl.b [ b ~ Li} for all i = 0, 1, 2 , . . . , r and Gi = GL,u, for i = 1, 2 . . . . . r, where GL,,, denotes the stabilizer of L~ui and u~:= L~ °. Of course, if H := L~, then H and G~ are subgroups of G. We will

prove that R = {G1, G2,. •. , Gr} is a semi-regular H-(s, klAI, ).1, 22)-partition of

G. If b' denotes fl,b w e have (L'i) ul ~ Gi because Gi = { f~ .b laeA and b ~ (u a)- 1Liui}. If b' ~ L'i, then A b'¢' = {fa,b* [ b* = (u? 1)"(b- 1)"bul and a ~ A}. But (b- 1)~b ~ Li because b ~ L~ and Li, by (4), is invariant under A. So f~,b* ~ Gi and we get A b'u' ~_ G~ for every b' ~ L'~. I t follows that (L'i)""w (Ub,eL,, Ab'u~) ~ Gv Conversely, using the fact that A is f.p.f., we obtain the inverse inclusion and so we have

(7) Gi = (L'i)u; w A v~; • b i

Now, ifb~ ~ b 2 and bx, b2~K, then A b " n A v~ = {1}. In fact, f , , b e A b l n A bi if

and only if b=(b?l )"b l =(b21)"b2 . Hence (b2b~-l)" = b2b? 1 with b2b~X# 1; this implies that a = 1 since A is f.p.f., and so b = 1 too. Therefore we have, by

(7)

IG~[--lEvi + ILi I ( IAI-1)=k[AI for all i = 1, 2 , . . . , r .

Again by (7), we infer f rom G~ = G s and i # j that (L'i) u' = (L)) ") and so L~ '~ = L~ ~ and L~u~ = L y j , a contradic t ion with p roper ty (3). It follows that the

componen t s of R are all distinct. Let x e G - H. If x e K' , then x = fl.b with b 6 Lo and so, by (1), b is on exactly 22 componen t s of [ul- 1Liui[ i = 1, 2 , . . . , r], and by (7) x belongs to exactly 22 componen t s of R. If x ¢ K' , there only exists

one element b' of K' such that x e A v. But, by (2), b is contained in exactly 22 componen t s of [L~uil i = 1, 2 , . . . , r]. So x is again contained in exactly 22 componen t s o fR . I f x e H - { 1 } , then x = f~,b with b e L o - { 1 } . It follows by

(1) that there exist exactly ).~ componen t s of [u[~L~u~[i= 1,2 . . . . , r ] containing b and so, by (7), x is contained in ).~ componen t s of R. Fo r i = 1 , 2 , . . . , r ,

[ G i ~ HI = ]( L'iff' ~ L'o] = ]uF X Liui ~ Lo];

hence, if k := klA], we have again by (1) that IGi~H[ = sklA]/(IK[ [A[) = sk/lG[, f rom which it follows that R is semi-regular and has the requested para- meters. This completes the proof.

292 A N T O N I N O G I O R G I O S P E R A

4. C O N S T R U C T I O N OF T D D ' S W H I C H A D M I T A F R O B E N I U S G R O U P

AS T H E I R T R A N S L A T I O N G R O U P

Let V(n, q) be a vector space over a field F with q elements, q a prime power. If

H is an / -d imens iona l subspace of V ( 1 < / < n - l ) and p={V1, V2 . . . . , V~} is

the set of all m-dimensional subspaces of V which meet H in some t-

dimensional subspace, where t = l + m - n > 0 and m < n, then p is an H-

(s, k, 21, 22)-partition of V with s = qZ and k -- qm. Moreover , the associated

(s, k, 21, 22)-TDD , D(p), is semi-regular. In fact, since t = l + m - - n > O, v = q"

divides s k = qZ+m and sk/v = q t = I ~ n H I , so D(p) is semi-regular by (ii) of Proposi t ion 2.2. Remark that it is possible to determine 21 and 22 from

rk =/)'~2 and (iii) of Propos i t ion 2.2, because r is known (see [5, Th. 3.1.2]) and

so are k, v and s. Now, let K be a hyperplane of V containing H and consider

the multiset ~ / = [ L o , L 1 , . . . , L r ] where L i = Vic~K for i = 1 , 2 . . . . . r and Lo = H . We will define a map ~ a : d - { L 0 } - - , L o and will consider a fixed

point free au tomorph i sm group A of K in such a way as to obtain a

generalized A-admissible triad (K, d , ~) with parameters s, k/q, 21 and 22. In

the meant ime we observe that ILol = IH[ = qZ and ILi l = I Vd IKI/I V~ + KI = q'~-1 for every i = l , 2 , . . . , r , since V I I + K = V (if V ~ + K = K , then

I Vz + HI = qm +l-t = q, implies K = l~, a contradiction). Let x be an element of

K - - L o. Since p is an H-(s, k, 21, 22)-partition of V, x is contained in exactly 22

components V h, Vii . . . . . . V~z2 of p. So x is contained in at least 22 components of d - - { L o } . But if x ~ L j = V j n K , then x ~ V 2. So Vj=V/~ for some

1 ~< e ~< 22. Therefore x is in exactly 22 components of ~ ' - {Lo}. In the same

way we can prove that each element of L o - { 0 } is contained in exactly 21

components of ~ ' - {Lo}. Moreover ,

[Lo n Li [ = IH n V~ n KI -- I n n ~ l = q' = qZ + m- ~ = q~ qm -1/lKi"

Thus proper ty (1) of the definition of a generalized A-admissible triad holds.

N o w , let Vo be a fixed vector of V - K . If x ~ K, Vo + x ¢ H because v o + x ¢ K

and H ~ K. So there exists 22 components of p containing v o + x. Consider the incidence structure A = (~, ~ ) with ~ = K and

= { L ~ + u l i = 1, 2 . . . . . r, u ~ K and U+Vo~ V~}.

Note that ~ is well defined because if x e L ~ + u ~ , then x + v o = l~ + u + Vo ~ l~ + V~ = V~ since l~ ~ Li ~_ V~. It is not difficult to show that A

is a tactical configurat ion with parameters v ' = q " - 1 k ' = qm-1 and r ' = 22. (Actually, we are able to prove that A is isomorphic to the tactical

configurat ion whose point set is the affine space P G ( n - 1 , q)r related to K and whose block set is { V~* I V~ ~ p}, where by V~* we denote the affine par t of

SEMI-REGULAR DIVISIBLE DESIGNS 293

the project ive subspace associa ted with V~.) So if b' denotes the number of

b locks of A, we have b ' = v ' r ' / k ' = v ' 2 2 / k ' . N o w let A' be the (qZ, k ' , 2 , 2 z ) -

divisible design, with v' points , associa ted with the Lo-(q ~, k', 21, 22)-part i t ion

I-LI, L 2 , . . . , L,]. Note tha t A' m a y get repea ted blocks because the compo-

nents of [L 1, L2 . . . . . L, ] are not necessari ly distinct. W e have shown above

that A' is semi-regular . So we have rk' = v'22 and, from b' = v'22/k', we ob ta in

b' = r. Tha t is, the tact ical conf igura t ion A has exact ly r blocks. Therefore w e

can choose sui table elements of K and write ~ = {Li + ui I i = 1, 2 , . . . , r}. But,

for each L i + u ~ g , ( L ~ + u l ) n L o v ~ ~ . So we can choose u~ in Lo, thus

ob ta in ing a m a p q~: ~¢ - {Lo) ~ Lo for which the p rope r ty (2) is t rue because,

as it has been proved, A is a tact ical conf igura t ion with r ' = 22. No te that if

Li = L j and i ~ j , then L~ + ui ~ L j + u j, since A has no repea ted blocks. So

p rope r ty (3) is p roved too. Final ly , in order to give an f.p.f, a u t o m o r p h i s m

group on K , we consider a non- t r iv ia l subgroup A of the mul t ip l ica t ive group

F* of F and we denote by A the group of au tomorph i sms of K of the form

x ~ x a with a E A. Of course, if d is the o rder of A, then d divides q - 1 .

Moreover , every componen t of ~¢ is invar ian t under ,4 and so p rope r ty (4) is

true. Therefore (K, d , q~) is a general ized A-admiss ib le t r iad on K with

pa ramete r s s = q~, k = q " - 1, 21 and ).2" Hence by P ropos i t i ons 3.2 and 2.3 we

get the fol lowing

T H E O R E M 4.1. L e t q, n, m and l be posit ive integers where q > 2 is a prime

power, m + l - n > O, m < n and l < n - l. I f d ~ l is a divisor o f q - l , then

there ex i s t s a semi-regular (s, k, 21, 22)- TDD, with s = qt, k = q m - l d, v = q" - l d

and suitable 21 and ).2, admit t ing a Frobenius group with abelian kernel as

translat ion group.

REFERENCES

1. Baer, R., 'Partitionen endlicher Gruppen', Math. Z. 75 (1961), 333-372. 2. Biliotti, M. and Micelli, C., 'On translation transversal designs', Ren. Sere. Mat. Univ.

Padova. 73 (1985), 217-229. 3. Bose, R. C. and Connor, W. S., 'Combinatorial properties of group divisible incomplete

block designs', Ann. Math. Star. 23 (1952), 367-383. 4. Herzer, A. and Schulz, R. H., 'Some new (s, k, 2)-translation transversal designs with non-

abelian translation group', J. Geometry 35 (1989), 87-96. 5. Hirschfeld, W. P., Projective Geometries over Finite Fields, Oxford, 1979. 6. Jungnickel, D., 'Transversal designs associated with Frobenius groups', J. Geometry 17

(1981), 140-154. 7. Jungnickel, D., 'On fiutomorphism groups of divisible designs', Canad. J. Math. 34 (1982),

257-297. 8. Schulz, R. H., 'Translation designs with 2 > 1 associated with Frobenius groups', Res. Math.

12 (1987), 401-410.

294 ANTONINO GIORGIO SPERA

9. Schulz, R. H., 'On the existence of generalized triads related to transversal designs', Ars. Combin. Ser B 25 (1988), 203-209.

10. Spera, A. G., 'Translation divisible designs', Arch. Math. 55 (1990), 507-515. 11. Zappa, G., 'Partizioni ed S-partizioni dei gruppi finiti'. Sym. Mat. 1st. Naz. Alta Mat. 1

(1968), 85-94.

A u t h o r ' s address:

A n t o n i n o G i o r g i o Spera,

Dip . di M a t e m a t i c a del l 'Univers i tY,

Via Archiraf i 34,

90123 P a l e r m o ,

I taly.

(Received, January 23, 1991; revised version, September 19, 1991)