The uniformly 3-homogeneous subsets of PGL\((2,q)\)
A set \(S\) of \(\text{Sym}_n\) is \(\mu\)-uniformly \(t\)-homogeneous if for any pair \(A\), \(B\) of unordered \(t\)-subsets, the same number \(\mu\neq 0\) of permutations in \(S\) carry \(A\) into \(B\). If the parameter \(\mu\) is not specified, we speak of a uniformly \(t\)-homogeneous set of permutations. Theorem 1. Let \(S\) be a uniformly 3-homogeneous proper subset of the group \(\text{PGL}(2, q)\), \(q\geq 4\). Then one of the following holds: (i) \(S= \text{PSL}(2,q)\) or \(S= \text{PGL}(2, q)- \text{PSL}(2, q)\), \(q\equiv 3\pmod 4\); (ii) \(q\in \{5, 7, 8\}\) and \(S\) is 3-uniformly 3- homogeneous. The 3-uniformly 3-homogeneous proper subset of \(\text{PGL}(2, q)\) for \(q= 5, 8\) has already been constructed.
scientific article; zbMATH DE number 754557
scientific article