On Permutation Groups of Prime Degree and Related Classes of Groups

Abstract

The transitive permutation groups of prime degree p appear as the Galois groups of the irreducible algebraic equations f(x) = 0 of degree p. This is the reason that these groups have been the subject of a large number of investigations.' However, only few results of a general nature have been obtained. In the present paper, the theory of group representations2 will be applied in order to derive some new theorems concerning the structure of these groups. Actually, the method can be used for the study of a wider class of groups, viz. the groups 5 of finite order g which have the following property: (*) The group (M contains elements P of prime order p which commute only with their own powers Pi. It is clear that transitive permutation groups of degree p have the property (*). Secondly, the doubly transitive permutation groups of degree p 1 are of this type.3 A third example is furnished by the irreducible linear groups in a p-dimensional vector space whose center consists of the unit element only, in particular by the simple linear irreducible groups in p dimensions (cf. section 7). It is easily seen (section 1) that the order g of a group (M with the property (*) is of the form