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# Basic Conditions

In this section we investigate some necessary conditions that have to hold if there exists a - design with automorphism group . So we throughout assume to be a point set of points. A -set simply means a subset of size of .

We start using only some divisibility conditions.

Theorem 1: If - is an admissible parameter set for a -design then is even. If in addition for some prime then 3 divides and 5 does not divide .

Proof By double counting the number of blocks of a - design is always

where .

Because a -design is also a -design with the same number of blocks the corresponding formula for yields the . In case and then such that is even.

Continuing with in the same way then such that if then 3 does not divide and hence has to divide The case is clearly impossible.

Continuing in this way we obtain from the formula for that 5 is a divisor of such that 5 does not divide .

We get some sharper results if we assume as a group of automorphisms. Our main tool will be to analyse orbits of on subsets of the point set by looking at the set stabilizer of such a subset.

We introduce some general notations. We assume that a finite group acts on a finite set of points. For a subset of the length of the orbit under is

Orbits on -sets and -sets for are related by some numbers which are important for the construction of -designs.

We define

and

Then we obtain a general formula already used by Alltop in 1965, [1].

Alltop's Lemma 2: Let a group act on a set and let be a -set and a -set of . Then

This follows easily by doubly counting

The group acts -homogeneously if it is transitive on the -sets.

Corollary 3: Let act -homogeneously in Alltop's Lemma. Then

Proof. Substituting , , and in the equation and cancelling yields the claimed result.

In the special case of and we obtain that divides 10. In particular, if 5 does not divide this implies that divides 2.

We will construct orbit representatives by constructing -sets invariant under a prescribed subgroup . From these we first have to remove those invariant under a larger group and then have to decide whether they lie in the same orbit. This can be done using the next Lemma [15].

Lemma 4 Let a group act on a set and let be two t-sets having the same stabilizer in . Let for some . Then .

Proof. We have

such that .

If 5 divides then 5 divides by Theorem 1. We know from group theory [10] that then there exists only one conjugacy class of subgroups of order in . Such a subgroup is a dihedral group with a cyclic normal subgroup of order . A subgroup of order 5 of then has orbits of length 5 and 1 only and . The orbits of length 5 form one orbit of on 5-element subsets. Thus, by Lemma 4, there is only one orbit of on 5-sets with a stabilizer of order 5. Let us assume to be a - design admitting . Then an orbit of of size 5 is contained in exactly one block of . This block is of size 6 and has to be invariant under . So, consists of and an additional fixed point of . We have to choose one of the two fixed points of to determine the first orbit of that has to belong to the design . The two choices lead to isomorphic solutions, since maps one of the orbits onto the other one.

We now investigate other orbits of 5-sets and 6-sets.

Theorem 5: Let and a 5-set whose stabilizer in has an order different from 5. Then For the block containing of a - design admitting we have that divides .

Proof. A subgroup leaving a 5-set invariant acts on . No elements of different from the identity has more than 2 fixed points. So, the orbits on must be of type (5), (4,1), (3,2), (3,1,1), (2,2,1). By assumption (5) does not occur. Since 2 does not divide , no element of order 2 of has any fixed point. So, (4,1), (2,2,1) do nor occur. By Corollary 3, no element of order 3 fixes . Therefore, and do not occur. So, The stabilizer of a 6-set then has orbits of length on the 5-sets contained in . So, where is the number of orbits.

We conclude that all stabilizers of blocks of a Steiner 5-system - with automorphism group must have orders in .

We want to derive some conditions on the numbers of blocks which lie in orbits of a fixed length.

Let be the number of orbits of size of on the blocks of the design. Then

From the general formula for the number of blocks in a -design we obtain in this case

Then,

has to be inserted into the equation.

After cancelling we obtain

This equation can be reduced modulo some primes to get short relations. Reducing modulo 5 gives our already known result that if 5 divides because also .

We consider possibilities of prescribing certain choices of stabilizers. If only one of all is assumed to be non-zero then we obtain the following results.

Theorem 6 Let a - design with automorphism group partition into - designs with automorphism group consisting of only one orbit of of length each. Then either and or and or and .

Proof. We only have to consider the cases . The case is already contained in [5]. So let now . Then

We write this as

and reduce modulo 4,5, and 9. Since is even, 4 divides . We have the solutions such that we have four combinations modulo 180 by the Chinese Remainder Theorem. The solutions and would imply that 3 divides . So, only the claimed cases remain for . There is one further case for where each of the only two existing orbits of 6-sets with a stabilizer of order 5 is already a - design. That is the well known small Witt design with the parameters -. In all other cases the number of orbits of length needed to form a - design is larger than the existing number of orbits on 6-sets with a stabilizer of order . To get this number of required orbits the number of bocks

of the 5-design has to be divided by . This number is . We will show in Theorem 7 that there exist only orbits with stabilizer and orbits with a stabilizer of order 6. In both cases there are no solutions.

One can discuss further more complicated cases of selecting several stabilizers with the same methods. We will give some experimental results on existing Steiner systems in such situations below.

Next: Subgroups of order up Up: Partitioned Steiner 5-Designs Previous: Introduction
N.N. 2002-02-25

University of Bayreuth -