Permutation groups of minimal movement

摘要

Let G be a permutation group on a set Ω with no fixed points in Ω and let m be a positive integer. Then we define the movement of G as, $m: = {text{move}}(G): =,sup _Gamma { |Gamma ^g backslash Gamma ||g in G} .$ Let p be a prime, p ≧ 5, and let move (G) = m. We show that if G is not a 2-group and p is the least odd prime dividing |G|, then n :=|Ω| ≦ 4m − p or n = 4m − p + 2. Moreover, the groups G attaining the maximum bound will be classified.