Transitive permutation groups with elements of movement m or m − 2

Abstract

Let G be a permutation group on a set \Omega with no fixed points in \Omega and let m be a positive integer. 
 If for each subset \Gamma of \Omegga the size |\Gamma^g \ Gamma| is bounded, for g in G, we define the 
 movement of g as the max |\Gamma^g \ Gamma| over all subsets \Gamma of \Omega, and the movement 
 of G is defined as the maximum of move(g) over all non-identity elements of g in G. In this paper we classify all 
 transitive permutation groups with bounded movement equal tomthat are not a 2-group, but in which every 
 non-identity element has movement m or m − 2.