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

Part of: Representation theory of groups Permutation groups

Published online by Cambridge University Press:  27 May 2019

MARK WILDON
MARK WILDON*
Affiliation:
Dept. of Mathematics, Royal Holloway, University of London, Egham Hill, Egham Tw20 OEX, U.K.
*
e-mail: mark.wildon@rhul.ac.uk
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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 paper 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.

MSC classification

Primary: 20B05: General theory for finite groups
Secondary: 20B15: Primitive groups 20C15: Ordinary representations and characters
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 613 - 633
Copyright
Copyright © Cambridge Philosophical Society 2019

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