Transitive Permutation Groups of Prime Degree

Abstract

This paper is intended as a survey of what is now known about transitive permutation groups of prime degree. The topic arises from early work on the theory of equations. Over 200 years ago Lagrange was led to an interest in irreducible polynomial equations of prime degree by showing1 that if every such equation were soluble in terms of root-extraction then polynomial equations of arbitrary degree would be. Even after Abel and Galois had shown that such solutions are impossible in general, Galois still devoted a good proportion of his work to equations of prime degree. It was of course he who emphasised the groups involved. Several 19th century mathematicians, notably Mathieu and Jordan, continued the work of Galois and provided foundations for the rich material that has been published since 1900. At present the problems concerning groups of prime degree remain near the centre of permutation group theory, retaining their interest partly as tests of the power and scope of techniques of finite group theory, partly as being typical of a range of similar problems concerning groups of degrees kp (with k < p) and p m where p is prime.KeywordsSimple GroupPermutation GroupTransitive GroupFinite Simple GroupPermutation RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.