On groups with no regular orbits on the set of subsets
Let G be a permutation group on a finite set \(\Omega\) of size n. Then G acts naturally on the set P(\(\Omega)\) of all subsets of \(\Omega\). In the present paper the following theorem is proved. Let G be primitive on \(\Omega\) of degree n, G neither alternating nor symmetric. Then, with only finitely many exceptions, G has a regular orbit on P(\(\Omega)\). The proof is based on the fact that if G is transitive of degree n then bm\(\geq n\), where m is the minimal degree of G on \(\Omega\) and b is the base size of G on \(\Omega\), and on order bounds for G obtained using the classification of finite simple groups [cf. \textit{P. J. Cameron}, Bull. Lond. Math. Soc. 13, 1-22 (1981; Zbl 0463.20003)]. Without the classification the theorem can be proved for 2-transitive G using \textit{L. Babai}'s bounds [Invent. Math. 65, 473-484 (1982; Zbl 0478.20002)]. It remains an open question whether a more elementary proof of the theorem can be given for simply primitive groups. The theorem implies some interesting consequences on some recent work of \textit{N. Inglis} [Arch. Math. 43, 297-300 (1984; Zbl 0545.20001)] and Siemons and Wagner (to appear).
scientific article; zbMATH DE number 3918627