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

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Abstract

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.
Original languageEnglish
Pages (from-to)1-21
Number of pages21
JournalMathematical Proceedings of the Cambridge Philosophical Society
Early online date27 May 2019
DOIs
Publication statusE-pub ahead of print - 27 May 2019

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