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algebras associated with the group of automorphisms of 2-designs. By ... f2, each of cardinality k; that is, I~[ = b and IBI = k for all B ~ ~. The pair D = (f2, N).
Arch. Math., Vol. 60, 247 251 (1993)

0003-889X/93/6003-0247 $ 2.50/0 9 ~993 Birkhfiuser Verlag, Basel

The group of automorphisms of non-associative commutative algebras associated with the group of automorphisms of 2-designs By M. A. SHAHABIand A. S. JANFADA

1. Introduction. Let f2 be a finite set with If21 = v, and ~ is a collection of b subsets of f2, each of cardinality k; that is, I~[ = b and IBI = k for all B ~ ~ . The pair D = (f2, N) is called a 2-design if each pair of distinct elements of f2 occurs in exactly one element of +~. Elements of ~ and ~ are called points and blocks, respectively. It is proved that each point occurs in the same, call r, blocks. The constants v, b, k and r are called the p a r a m e ters of D and it is proved that they satisfy in the following relations:

bk = vr,

r(k - 1) = v -

1.

An a u t o m o r p h i s m of D is defined as a p e r m u t a t i o n on f2 which induce a p e r m u t a t i o n on ~ , and the group of a u t o m o r p h i s m s of D is denoted by Aut (D). Let D = (f2, ~ ) be a 2-design with parameters v, b, k and t+, k > 3. Put ~2 = {0, 1. . . . . n}. Let i,j ~ ~ with i =4=j, assume that G is a subgroup of Aut (D) acting 2-transitively on g2 and suppose that {i,j}, B ( i , j } - {i,j} and f2 - B ( i , j ) are the orbits of G,,;(the global stabilizer of {i,j} in G) on f2, where B ( i , j ) denotes the unique block containing i a n d j . Let A be the commutative (non-associative) G-invariant algebra over a field K of characteristic + 2. Then by determination of the general structure of commutative (non-associative) algebras associated with doubly transitive p e r m u t a t i o n groups G by H a r a d a , Theorem 2.4 of [3], A has basis {x 1, x 2 . . . . . xn} and

x l x i = dxi,

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