Izumi Miyamoto

Classification of association schemes of small order

This paper enumerates the isomorphism classes of association schemes of order 20-28 by using the computer. It also classifies all the association schemes of order 24-28 whether their automorphism groups are transitive or not, and whether they are group case or not. Some more properties of the obtained association schemes are computed.

Deficiency zero presentations for certain perfect groups

In this paper we give two generator, two relation presentations for the following perfect groups which were not previously known to have deficiency zero: SL(2 , 32), SL(2 , 64), SL(2 , 27), SL(2 , 49), Â 7 , Ŝz(8), SL(2 , 5) × SL(2 , 5), SL(2 , 5) × SL(2 , 25). We also give two generator, two relation presentations for three other finite perfect groups, two having SL(2 , 7) as an image and one having SL(2 , 5) as an image. We also discuss presentations for certain other perfect groups which were known to have deficiency zero and find some neat new presentations for them.

Computing normalizers of permutation groups efficiently using isomorphisms of association schemes

This note presents an algorithm to speed up the computation of normalizers of permutation groups. It is an application of computation of isomorphisms of association schemes.

Computing Isomorphisms of Association Schemes and its Application

Isomorphisms of association schemes are isomorphisms of edge-labeled regular graphs permitting to permute the labels. We give an algorithm computing isomorphisms of association schemes using their algebraic property. We also study an application of computing isomorphisms of association schemes to computing normalizers of permutation groups.

A computation of some multiply homogeneous superschemes from transitive permutation groups

Let <i>G</i> be a doubly transitive permutation group on a set <i>X</i>. A doubly homogeneous superscheme is formed by the orbits on the set of triples of <i>X</i> of <i>G</i>. Let α be a point of aset <i>X</i> and let <i>H</i> be a transitive group on <i>X</i>\{α}. Then from the combinatorial structure of the superscheme formed by the orbits of <i>H</i> on <i>X</i>³,we may construct some doubly homogeneous superschemes on <i>X</i>. We will give a general algorithm to compute such superschemes and show how to implement it practically. In particular if <i>H</i> = <i>G</i>_α, thestabilizer of α in <i>G</i>, then we can construct a superscheme of which automorphism group is <i>G</i> in the cases of moderate size. Furthermore, even if <i>H</i> is not a stabilizer of a doubly transitive group, we can consider some orbit-like sets of a doubly homogeneous superscheme. We see whether such sets form a design in some cases. As a related combinatorial algorithm we have developed a program to compute the automorphism group of a superscheme which is a kind of a labeled hyper graph.

A combinatorial approach to doubly transitive permutation groups

Let G be a doubly but not triply transitive group on a set X. We give an algorithm to construct the orbits of G acting on XxXxX by combining those of its stabilizer H of a point of X If the group H is given first, we compute the orbits of its transitive extension G, if it exists. We apply our algorithm to G=PSL(m,q) and Sp(2m,2), m>=3, successfully. We go forward to compute the transitive extension of G itself. In our construction we use a superscheme defined by the orbits of H on XxXxX and do not use group elements.

Computation of isomorphisms of coherent configurations

A program computing isomorphisms between association schemes was applied to speed up the computation of normalizers of permutation groups. A transitive permutation group forms an association scheme while a permutation group which may not be transitive forms a coherent configuration. In this paper we discuss the extension of the above program to compute isomorphisms between coherent configurations and show some typical examples of computing normalizers.

An improvement of GAP normalizer function for permutation groups

In GAP system it takes unreasonably long time to compute the normalizers of some permutation groups, even though they are of small degree. The author gave an algorithm in [7, 8] to compute the normalizers of permutation groups and particularly it worked smoothly for transitive groups of degree up to 22. In 1999 GAP version 4 was released. Since then the GAP system has been improved and in 2004 GAP4r4 had a special function to compute the normalizers in the symmetric groups but it still has difficulties in computing the normalizers of some permutation groups. It has been also found that the authors algorithm in [7, 8] has difficulties in some groups of small degree but larger than 22. So the author will give two new programs improving the computation of normalizers of transitive permutation groups in the symmetric groups. One of them works comparatively smoothly for the transitive groups of degree up to 30.

A construction of designs on n+1 points from multiply homogeneous permutation groups of degree n

A simple criterion to construct a t-design on n+1 points from a t-ply homogeneous permutation group of degree n using some orbits of the group is obtained. The design is not simple in general. Applying the criterion to 2-dimensional projective special linear groups PSL(2,q) for q>=19 acting on the projective line of q+1 points, simple 3-(q+2,12(q-1),124(q-1)(q-3)(q-5)) designs are obtained if q=3 or 7 (mod 12), and simple 3-(q+2,12(q+1),18(q-1)^2(q-3)) designs are obtained if q=3 (mod 4).

A Construction of Designs from PSL(2,q) and PGL(2,q), q=1 mod 6, on q+2 Points

Let G=PSL(2,q) or PGL(2,q). We consider the action of G on the projective line together with one additional point, which is fixed by G. Assume q≡1 mod 6 and set \( \lambda {}_q = \frac{1}{{24}}\left( {q - 1} \right)\left( {q - 3} \right)\left( {q - 5} \right). \) We construct \( 3 - \left( {q + 2,\frac{1}{2}\left( {q - 1} \right),{\lambda _q}} \right) \) designs admitting PSL(2,q) as their automorphisms, if q≡3 mod 4. We also construct \( 3 - \left( {q + 2,\frac{1}{2}\left( {q - 1} \right),2{\lambda _q}} \right) \) designs admitting PGL(2,q) as their automorphisms. These designs may not be simple.