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A group K is said to be a B-group if every permutation group containing K asa regular subgroup is either imprimitive or 2-transitive. In the second editionof his influential textbook on finite groups, Burnside published a proof thatcyclic groups of composite prime-power degree are B-groups. Ten years later in1921 he published a proof that every abelian group of composite degree is aB-group. Both proofs are character-theoretic and both have serious flaws.Indeed, the second result is false. In this note we explain these flaws andprove that every cyclic group of composite order is a B-group, using onlyBurnside's character-theoretic methods. We also survey the related literature,prove some new results on B-groups of prime-power order, state two related openproblems and present some new computational data.

Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside