The Double Transitivity of a Class of Permutation Groups

Abstract

Certain finite groups H do not occur as a regular subgroup of a uniprimitive (primitive but not doubly transitive) group G. If such a group H occurs as a regular subgroup of a primitive group G, it follows that G is doubly transitive. Such groups H are called B-groups (8) since the first example was given by Burnside (1, p. 343), who showed that a cyclic p-group of order greater than p has this property (and is therefore a B-group in our terminology).Burnside conjectured that all abelian groups are B-groups. A class of counterexamples to this conjecture due to W. A. Manning was given by Dorothy Manning in 1936 (3).