A combinatorial approach to doubly transitive permutation groups

Known doubly transitive permutation groups are well studied and the classification of all doubly transitive groups has been done by applying the classification of the finite simple groups. Readers may refer to [6]. This may mean that it is not still sufficient to study doubly transitive groups as permutation groups. Typical arguments on doubly transitive groups are seen in the book[3]. In the present paper we will give a combinatorial approach to doubly transitive groups. We use a combinatorial structure called a superscheme defined by the stabilizer of a point in a doubly transitive group and try to construct the superscheme defined by the given doubly transitive group. So if a superscheme defined by a transitive group is given first, our algorithm will construct the superscheme defined by the expected transitive extension of the given group. We do not use any group elements in this construction. Then if no superscheme is constructed, we can conclude that there is no transitive extension. If a superscheme is constructed, its automorphism group can be expected to be closely related to the transitive extension. However we will not compute the automorphism group, since we can guess what doubly transitive group it is. We will show how our algorithm works in the case of projective special linear groups PSL(m, q), m ≥ 3. Our algorithm also works successfully in the case of symplectic groups Sp(2m, 2), m ≥ 3, over GF (2) acting on the cosets by O(2m, 2) and O−(2m, 2). In PSL-cases the automorphism groups of the obtained superschemes are usually imagined to be PGL. We also compute the transitive extensions of these groups itself. Our algorithm shows that they do not have transitive extensions except that PSL(m, 2) and PSL(3, 4), which are known to give AGL(m, 2) and the Mathieu group M22 of degree 22, and so on, we can consider the extensions of the obtained groups. Computer experiments suggest that our algorithm can be applied to any doubly or more transitive groups. We note that an obtained superscheme may not have a doubly transitive automorphism group. Association schemes are often used to study permutation groups. A doubly