On minimal p-degrees in 2-transitive permutation groups

Abstract

Suppose G is a transitive permutation group on a finite set \(\mit\Omega \) of n points and let p be a prime divisor of \(|G|\). The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted mp (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that \(m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega |\) holds, if \({\rm Alt}(\mit\Omega ) \nleqq G \).¶Also all groups G (and prime divisors p of \(|G|\)) for which \(m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega |\) are identified.