A construction and characterization of new sharply 3-transitive permutation sets contained in \(P\varGamma L(2,K)\)
A set \(S\) of permutations on a finite set \(\Omega\) is said to be sharply \(k\)-transitive if for any two \(k\)-tuples \((x_1,\cdots,x_k)\) and \((y_1,\cdots,y_k)\) of distinct elements in \(\Omega\), there exists a unique permutation \(\alpha\in S\) such that \(x_i^{\alpha}=y_i\) for \(i=1,\cdots,k\). Sharply \(k\)-transitive permutation sets have various applications in the construction of combinatorial and geometric structures. The main contribution of this paper is a charaterization of the sharply 3-transitive permutation groups and sets in \(P\Gamma L(2,K)\) whose permutations are defined from two automorphisms of the finite field \(K\). It covers all known sharply 3-transitive permutation groups and sets on \(K\cup \{\infty\}\), and it also provides new constructions.
scientific article; zbMATH DE number 7451587