Symmetric designs for (71,15,3) with the semi-standard automorphism group G E8 · F21 have been investigated. There were constructed exactly three nonisomorphic designs, two of them with an involutory elation

Mirjana Garapić

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GLASNIK MATEMATIČKIVol. 35(55)(2000), 211 – 214SOME SYMMETRIC (71, 15, 3) DESIGNS WITH ANINVOLUTORY ELATIONMirjana GarapićUniversity of Zagreb, CroatiaAbstract. Symmetric designs for (71, 15, 3) with the semi-standardautomorphism group G ∼= E8 · F21 have been investigated. There wereconstructed exactly three nonisomorphic designs, two of them with an in-volutory elation.1. IntroductionIn [1], we have proved that exactly six nonisomorphic triplanes for(71, 15, 3) exists having the automorphism group G ∼= E8 · F21 acting non-semistandardly. The group G is a faithful extension of an elementary abeliangroup E8 of order 8 by a Frobenius group of order 21. Here we shall con-fine our attention to the semi-standard automorphism group situation. Weuse the method from [2] and this group G to ”construct” the triplanes for(71, 15, 3) under this assumption. We have found that there exist triplaneswith these parameters nonisomorphic to the triplanes in [1], admitting anelation of order 2 and also another class of triplanes without that property.Hence, the property of admitting an involutory elation yields a new approachto the construction and the classification of symmetric design for (71, 15, 3).2. Notation and basic definitionsSuppose G is an authomorphism group of a symmetric D = (v, k, λ)design. Actually, G has orbits of lengths a1, a2, . . . , ac on points and orbits oflengths b1, b2, . . . , bc on blocks of D respectively.2000 Mathematics Subject Classification. 05B05.Key words and phrases. symmetric design, triplane, automorphism group, fixed point(block).211212 MIRJANA GARAPIĆdefinition 1. An automorphism group G is said to be semi-standard if,after possibly renumbering the orbits we have ai ≤ aj and bi ≤ bj for i ≤ jand if ai = bi for all i = 1, 2, . . . , c.We also say that a group G acts semistandardly. In the oposite case, wesay G acts non-semistandardly.definition 2. Let α 6= 1 be an automorphism on a symmetric 2−(v, k, λ)design. We say that α is an elation if there exists an α-fixed block b containingexactly k α-fixed points and if α fixes a set of k blocks having exactly one α-fixed point of b in common.Notation and other definitions can be found in [1].3. Results of the investigation in thesemi-standard group situationNow, suppose the group G ∼= E8 · F21 is an automorphism group of asymmetric 2 − (71, 15, 3) design. This group G is generated by the set ofthree elements of order two, three and seven respectively. (For details ofthe action of generators see [1].) It was proved in [1] that an involutionhad seven or fifteen points, an automorphism of order three fixed five pointsand an automorphism of order seven fixed exactly one point. In the casewhere G acts semistandardly it was proved that G had three orbits on thepoints(blocks) having lengths 1, 14, 56. Here we have constructed symmetricdesigns D5, D6 and D7, all of them nonisomorphic. (See the method in [2].)Without loss of generality the constructed designs have the following blockorbit representatives:The first block orbit representative of D5, D6 and D7:11 21 22 23 24 25 26 27 28 29 210 211 212 213 214The second block orbit representative of D5, D6 and D7:11 21 22 38 39 311 318 319 321 325 333 338 345 351 353The third block orbit representatives forD5 : 25 27 28 38 39 311 315 316 326 327 330 335 348 352 355D6 : 25 27 28 38 39 311 315 327 328 332 337 342 346 349 352D7 : 25 27 28 310 312 313 316 325 326 330 333 335 348 351 355Here the intersection numbers of all triples of blocks for D5,D6 and D7 havebeen calculated but the constructed designs D6 and D7 do not differ in thischaracteristic. Hence we have used the method of spreads (see [1]) to provethe nonisomorphism of designsD5, D6 andD7. We proved that all of them aremutually nonisomorphic as well as nonisomorphic to the designs constructedSOME SYMMETRIC (71, 15, 3) DESIGNS WITH AN INVOLUTORY ELATION 213in [1]. Compare the results of investigation in the Table 1 with the results in[1].Table 1: Di b∗ ( r1, r2, r3 ) ( s1, s2, s3 )D5 1 (35, 63, 69) (56, 52, 100)D6 1 (35, 51, 63) (56, 160, 86)D7 1 (35, 51, 63) (56, 160, 70)Here, b∗ is the number of special blocks of a design Di, ri means thenumber of blocks of a reduced structure Ib/R where b is an element of thei-th orbit of blocks of Di and si means the number of spreads with respect tothe block b in the i-th orbit of blocks of Di.It was interesting to investigate the acting of involution f = ρ∗ fixingfifteen points of the constructed designsD5 ,D6 and D7. (See the permutationρ∗ in [1]). Here we have proved that f was not an automorphism on D5 butf acted on D6 and D7 as an elation.Now the first orbit of blocks of D6 and D7 contains just one G-fixedblock containing exactly fifteen f -fixed points. The second orbit consists offourteen blocks which all have the point 11 in common. We find that theblocks incident with the point 11 are all the f -fixed blocks. Here a pair ofpoints of these blocks is either fixed or interchanged by f . The third orbitof blocks of D6 and D7 contains further fifty-six blocks of these designs. Byverification it turns out that this permutation f breaks up this orbit of blocksin twenty-eight pairs of blocks interchanged by f for both D6 and D7. Asone can see, the permutation f is an elation on the symmetric designs D6and D7 respectively. This automorphism f is different from, and commuteswith the automorphisms provided by the group G and it was found under thisassumption. We have seen that the permutation f was not an automorphismon D1, it acted as an automorphism on the symmetric designs D2 and D3(see in [1]) but it was not an elation on designs D2 and D3.We have investigated only the symmetric (71, 15, 3) designs with the au-tomorphism group G ∼= E8 ·F21. We did not check whether it was possible toproduce still more symmetric (71, 15, 3) designs with an involutory elation inanother case. Thus we have the following result:Theorem 3.1. There exist at least two non-isomorphic symmetric(71, 15, 3) designs admitting an involutory elation. Both of them are non-self-dual.References[1] M.Garapić, Triplanes for (71, 15, 3) admitting a solvable group of order 168 acting non-semistandardly, Glasnik Matematički 29(49)(1994), 17–24.[2] Z.Janko, Coset enumeration in groups and constructions of symmetric designs, Com-binatories 90(1992), 275–277.214 MIRJANA GARAPIĆZavod za matematiku, informatiku i nacrtnu geometrijuRudarsko-geološko-naftni fakultetPierottijeva 610000 ZagrebCroatiaReceived : 210.01.1999.Revised : 22.06.2000

Some symmetric (71,15,3) designs with an involutory elation

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