Certain finite groups H do not occur as a regular subgroup of a unirimitive (primitive but not doubly transitive) group G. If such a group i occurs as a regular subgroup of a primitive group G, it follows that G is doubly trans¬itive. Such groups H are called B-groups 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).nThis paper is a generalization of these results. Let H be abelian, P a Sylow p-subgroup of H and a an element of P of maximal order, pa. Let A be the cyclic group generated by a.
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