Naslov
Construction of block designs admitting an abelian automorphism group

Autori
Crnković, Dean ; Rukavina Sanja

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Conference on Recent Advances in Combinatorial Designs and Related Combinatorics Abstracts / - Atena, 2003, 10-11

Skup
International Conference on Recent Advances in Combinatorial Designs and Related Combinatorics

Mjesto i datum
Atena, Grčka, 07.07.2003. - 09.07.2003

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
block designs; automorphisms

Sažetak
Construction of block designs admitting an action of presumed automorphism group consists of two basic steps: 1. construction of orbit structures for the given automorphism group, 2. construction of block designs for the obtained orbit structures. Because of the large number of possibilities, it is often necessary to involve a computer in both steps of the construction. Let \( G \) be an abelian group. Then $ G= C_{1} \times C_{2} \times \ldots \times C_{n}$, where $C_1, \ldots , C_n$ are cyclic groups. For such a group $G$, the second step, called indexing, often lasts too long. Therefore, we use a principal series $1 \triangleleft C_{1} \triangleleft C_{1} \times C_{2} \triangleleft \ldots \triangleleft G$ of the group $G$ for the "refinement" of the orbit structures. The method of construction now consists of $n+1$ steps: Step 1 Construction of orbit structures for the group $G$. Step 2 Construction of the corresponding orbit structures for the subgroup $C_{1} \times C_{2} \times \ldots \times C_{{n-1}}$. . . . Step n Construction of the corresponding orbit structures for the subgroup $C_{1}$. Step (n+1) Indexing of the orbit structures for $C_{1}$, having in mind the action of the subgroups $C_{2}$, $C_{3}$, $\ldots$, $C_{n-1}$ and $C_{n}$ on the designs. This method could be used for construction of any incidence structure with an abelian automorphism group. REFERENCES [1] D. Crnkovi\'c, Symmetric (70, 24, 8) designs having $Frob_{21} \times Z_2$ as an automorphism group, Glas. Mat. Ser. III Vol. 34(54) (1999), 109--121. [2] D. Crnkovi\'c, S. Rukavina, Unique symmetric (66, 26, 10) design admitting an automorphism of order 55, Math. Commun. 6, No.1 (2001), 83--87. [3] Z. Janko, Coset Enumeration in Groups and Constructions of Symmetric Designs, Combinatorics '90 (1992), 275--277. [4] S. Rukavina, Some new triplanes of order twelve, Glas. Mat. Ser. III Vol. 36(56) (2001), 105-125.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA