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Transitive permutation groups with elements of movement \(m\) or \(m-1\). (English) Zbl 1289.20001
Summary: Let \(G\) be a permutation group on a set \(\Omega\) with no fixed point in \(\Omega\) and let \(m\) be a positive integer. If for each subset \(\Gamma\) of \(\Omega\) the size \(|\Gamma^g\setminus\Gamma|\) is bounded, for \(g\in G\), we define the movement of \(g\) as the matrix \(|\Gamma^g\setminus\Gamma|\) over all subsets \(\Gamma\) of \(\Omega\), and the movement of \(G\) is defined as the maximum of \(\text{move}(g)\) over all non-identity elements of \(g\in G\). In this paper we will classify all transitive permutation groups \(G\) with bounded movement equal to \(m\), such that \(G\) is not a 2-group but in which every non-identity element has the movement \(m\) or \(m-1\).
20B05 General theory for finite permutation groups
20B20 Multiply transitive finite groups