On the point stabilizers of transitive groups with non-self-paired suborbits of length 2

In [7] a characterization of transitive permutation groups having a non-self-paired suborbit of length 2 (with respect to which the corresponding orbital graph is connected) was obtained in terms of their point stabilizers. As a consequence, elementary abelian groups were proved to be the only possible abelian point stabilizers arising from such actions, and D8 was shown to be the only nonabelian group of order 8 with the same property. Constructions of such group actions with point stabilizers isomorphic to D8 or to Z h2 , h 1, were also given there. These results are extended here to include a more in depth analysis of the structure of point stabilizers of such group actions, resulting in a set of necessary conditions allowing us to obtain a restricted list of 19 possible candidates for point stabilizers of such group actions when the point stabilizers have order 2h, h 8. (For h 5, this list gives a complete classi cation of such point stabilizers.) Furthermore, a construction of a transitive permutation group action with a non-self-paired suborbit of length 2 and point stabilizer isomorphic to D8 Z h 3 2 is given for each h 3. 1991 MSC: 05C25, 20B25.

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