  A group K is said to be a B-group if every permutation group containing K as
a regular subgroup is either imprimitive or 2-transitive. In the second edition
of his influential textbook on finite groups, Burnside published a proof that
cyclic groups of composite prime-power degree are B-groups. Ten years later in
1921 he published a proof that every abelian group of composite degree is a
B-group. Both proofs are character-theoretic and both have serious flaws.
Indeed, the second result is false. In this note we explain these flaws and
prove that every cyclic group of composite order is a B-group, using only
Burnside's character-theoretic methods. We also survey the related literature,
prove some new results on B-groups of prime-power order, state two related open
problems and present some new computational data.
