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

## TL;DR

This paper examines historical errors in Burnside's proofs regarding B-groups, clarifies the correct results about cyclic groups of composite order, and explores related open problems with new computational insights.

## Contribution

It corrects and clarifies Burnside's flawed proofs, establishes that all cyclic groups of composite order are B-groups, and surveys related literature and open problems.

## Key findings

- All cyclic groups of composite order are B-groups.
- Burnside's original proofs contain serious flaws.
- New computational data on B-groups of prime-power order.

## 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.