Movement of intransitive permutation groups having maximum degree

Abstract

Let G be a permutation group on a set Ω with no fixed points in Ω, and m be a positive integer. Then the movement of G is defined as $move(G): = \mathop {\sup }\limits_\Gamma \left\{ {\left| {\Gamma ^g \backslash \Gamma } \right|\left| {g \in G} \right.} \right\}$. It was shown by Praeger that if move(G) = m, then |Ω| ≤ 3m+ t−1, where t is the number of G-orbits on Ω. In this paper, all intransitive permutation groups with degree 3m+t−1 which have maximum bound are classified. Indeed, a positive answer to her question that whether the upper bound |Ω| = 3m + t − 1 for |Ω| is sharp for every t > 1 is given.

Keywords

Intransitive permutation groups / Bounded movement / Orbit

References

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