Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside
Abstract
A group K is said to be a Bgroup if every permutation group containing K as a regular subgroup is either imprimitive or 2transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite primepower degree are Bgroups. Ten years later in 1921 he published a proof that every abelian group of composite degree is a Bgroup. Both proofs are charactertheoretic 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 Bgroup, using only Burnside's charactertheoretic methods. We also survey the related literature, prove some new results on Bgroups of primepower order, state two related open problems and present some new computational data.
 Publication:

arXiv eprints
 Pub Date:
 May 2017
 DOI:
 10.48550/arXiv.1705.07502
 arXiv:
 arXiv:1705.07502
 Bibcode:
 2017arXiv170507502W
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Representation Theory;
 20B05;
 secondary 20B15;
 20C20
 EPrint:
 24 pages