Johannes Siemons

Doubly homogeneous 2-( v, k, 1) designs

Abstract If D is a 2-(v, k, 1) design admitting a group G of automorphisms which acts doubly homogeneously but not doubly transitively on the points, we prove that v = pn for some prime p ≡ 3 (mod 4), n is odd and 1. (1) D is an affine space over a subfield of GF(pn) or 2. (2) D is a Netto system, k = 3 and p ≡ 7 (mod 12).

Regular Sets and Geometric Groups

If G is a permutation group acting on a set Ω, a subset Λ of Ω is called a regular set for G if the set-stabilizer of Λ in G is the identity subgroup. We show here that the projective and affine semi-linear groups acting in the natural way as permutation groups on their respective finite geometries, have, in general, for all finite dimensions and all finite fields, regular sets of points. The exceptions to this are found, and an extension of the results to infinite fields is discussed.

Error graphs and the reconstruction of elements in groups

Packing and covering problems for metric spaces, and graphs in particular, are of essential interest in combinatorics and coding theory. They are formulated in terms of metric balls of vertices. We consider a new problem in graph theory which is also based on the consideration of metric balls of vertices, but which is distinct from the traditional packing and covering problems. This problem is motivated by applications in information transmission when redundancy of messages is not sufficient for their exact reconstruction, and applications in computational biology when one wishes to restore an evolutionary process. It can be defined as the reconstruction, or identification, of an unknown vertex in a given graph from a minimal number of vertices (erroneous or distorted patterns) in a metric ball of a given radius r around the unknown vertex. For this problem it is required to find minimum restrictions for such a reconstruction to be possible and also to find efficient reconstruction algorithms under such minimal restrictions. In this paper we define error graphs and investigate their basic properties. A particular class of error graphs occurs when the vertices of the graph are the elements of a group, and when the path metric is determined by a suitable set of group elements. These are the undirected Cayley graphs. Of particular interest is the transposition Cayley graph on the symmetric group which occurs in connection with the analysis of transpositional mutations in molecular biology [P.A. Pevzner, Computational Molecular Biology: An Algorithmic Approach, MIT Press, Cambridge, MA, 2000; D. Sankoff, N. El-Mabrouk, Genome rearrangement, in: T. Jiang, T. Smith, Y. Xu, M.Q. Zhang (Eds.), Current Topics in Computational Molecular Biology, MIT Press, 2002]. We obtain a complete solution of the above problems for the transposition Cayley graph on the symmetric group.

The Modular Homology of Inclusion Maps and Group Actions

Let?be a finite set ofnelements,Ra ring of characteristicp>0 and denote byMktheR-module withk-element subsets of 0 as basis. The set inclusion map ?: Mk?Mk?1is the homomorphism which associates to ak-element subset 2 the sum ?(?)=?1+?2+?+?kof all its (k?1)-element subsets?i. In this paper we study the chainformula]arising from ?. We introduce the notion ofp-exactness for a sequence and show that any interval of (*) not includingMn/2orMn+1/2respectively, isp-exact for any primep>0. This result can be extended to various submodules and quotient modules, and we give general constructions for permutation groups on?of order not divisible byp. If an interval of (*) , or an equivalent sequence arising from a permutation group on?, does include the middle term then proper homologies can occur. In these cases we have determined all corresponding Betti numbers. A further application arep-rank formulae for orbit inclusion matrices.

Regular sets on the projective line

We show that if G is the group PΓL(2,q)(for q a prime-power) acting on the points of the projective line in the usual way, then for q>27 there is a set Λ of 5 points such that no non-trivial element of G fixes Λ.

Regular orbits of cyclic subgroups in permutation representations of certain simple groups

If H acts on ∆ then an H -orbit is regular if its cardinality is |H |. The alternating groups, already in their natural representation, do not have the property of the theorem, hence the exception. The other known simple groups with a doubly transitive permutation representation are PSL(n, q), Sp(2n,2) (two representations), U3(q), B2(q), G2(q) and a short list of sporadic examples which are reproduced in Section 5. If one assumes the completeness of the classification of finite simple groups then these are all doubly transitive representations of finite simple groups and the word known can be omitted in the theorem. In our paper [8] the Theorem 1.1 was proved for PSL(n, q). Here we consider the remaining doubly transitive groups. The same method can in

On finite permutation groups with the same orbits on unordered sets

By JOHANNES SIEMONS and ASCHER WAGNER 1. Introduction. A permutation group G acting on a set f2 induces a permutation group on the unordered sets of k distinct points. If H is another permutation group on f2 we shall write H ~ G if H and G have the same orbits on the unordered sets of k points. Bercov and Hobby [2] have shown that for infinite groups H k G implies H ,,~ G if I < k. In [9] we have shown that this result is also true for finite groups, with the obviously necessary condition that k < 89 f2l- In [9] it is also shown that for finite groups H 2 G implies that H and G are either both primitive or both imprimitive with the same blocks of imprimi- tivity. If H and G have the same orbits on all subsets of ~2 we shall say that H and G are orbit equivalent and write H ~ G. Orbit equivalence for groups acting on quite arbitrary f2 has been considered by Betten [3]; the main results concern intransitive groups. In this paper we shall be concerned only with orbit equivalence for finite groups. In this case, of course, H ~ G if, and only if, H k G for all k. Suppose that H ~ G and that L is a permutation group on a set A. Then the direct products H x L and G x L, acting naturally on f2 u A, are orbit equivalent and intransi- tive. Also, if L is transitive on A the wreath products H 2, L and G 2, L, acting naturally on the direct sum of [A h copies of f2, are orbit equivalent and imprimitive. This suggests that the basic situation to investigate is when G, hence also H, is primitive on f2. Without loss of generality we may assume that H c G since H ~ G implies H ~ (H, G). Our main result is the following theorem. Theorem A. Let K be a finite primitive permutation group on a set f2. Let H c K and H ,~ K. Suppose there exists aprime r dividing the order of K but not the order of H. Then only the following possibilities exist: H K I g21 r 2-sets 3-sets 4-sets (i) ~3 5;3 3 2 (i 0 C 5 Dlo 5 2 5; 5 (iii) A~ (5) S s 5 3 10 (iv) A x (8) FA~ (S) 8 3 28 (V) A 1 (8) 23 \\PSL 3 (2) 8 3 28 (vi)

On the reconstruction index of permutation groups: semiregular groups

Summary. The reconstruction index of all semiregular permutation groups is determined. We show that this index satisfies

Intertwining automorphisms in finite incidence structures

3 \leq \rho(G, \Omega) \leq 5

Efficient reconstruction of partitions

and we classify the groups in each case.