Extension of Multiply Transitive Permutation Sets

Abstract

Let G be a k-transitive permutation set on E and let E* = E∪{∞},∞ ∉ E; if G* is a (k: + 1)-transitive permutation set on E*, G* is said to be an extension of G whenever G ^*_∝ =G. In this work we deal with the problem of extending (sharply) k- transitive permutation sets into (sharply) (k + 1)-transitive permutation sets. In particular we give sufficient conditions for the extension of such sets; these conditions can be reduced to a unique one (which is a necessary condition too) whenever the considered set is a group. Furthermore we establish necessary and sufficient conditions for a sharply k- transitive permutation set (k ≥ 3) to be a group. Math. Subj. Class.: 20B20 Multiply finite transitive permutation groups 20B22 Multiply infinite transitive permutation groups