Burnside proved in 1901 that if is an odd prime then a permutation group containing a regular subgroup isomorphic to is either imprimitive or 2-transitive. His proof was an early applications of character theory to permutation groups. Groups with this property are now called B-groups.

Burnside attempted to generalize his 1901 result in two later papers: in 1911, he claimed a proof that is a B-group for any prime and any , and in 1921, he claimed a proof that all abelian groups, except for elementary abelian groups, are B-groups. The first claim is correct, but his proof has a serious gap. This error appears to have been unobserved (or, just possibly, observed but ignored, since the result was soon proved in another way using Schur’s theory of -rings) until 1994 when it was noted by Peter Neumann, whose explication may be found in his introduction to Burnside’s collected works. In 1995, Knapp extended Burnside’s argument to give a correct proof. Burnside’s second claim is simply false: for example, acts primitively on , and has a regular subgroup isomorphic to . In one of my current projects, I’ve simplified Knapp’s proof and adapted Burnside’s character-theoretic methods to show, more generally, that any cyclic group of composite order is a B-group.

The purpose of this post is to record some proofs omitted for reasons of space from the draft paper. This companion post has some notes on B-groups that may be of more general interest.

Let be a primitive th root of unity. Define

for . Define a subset of to be *null* if there exists and distinct for and such that mod for each and and

**Proposition 6.2** Let where is not divisible by . Let . Then

if and only if *either*

- is null;
*or* - where is a null set, the are distinct elements of and mod for each .

*Proof.* Since the minimum polynomial of is

we have . Since , we have . It follows that if is a null set. (For the second equality, note the contributions from the for fixed combine to give .) For (2) we have , and . This proves the ‘if’ direction.

Conversely, by Lemma 2.1 in the paper, is a union of some of some of the sets . There exists a unique subset of and unique for and unique for and such that

We have and . Therefore

has as a root. Since this polynomial has degree at most and the minimal polynomial of is , it follows that the coefficients are constant. Hence

for . If then and is null. Otherwise, taking the previous displayed equation mod we see that , and, moreover, the are constant for . (This holds even if .) Set

We have . Hence, and choosing in any way for such that , we see that is the union of , the sets for and a null set. .

**Proposition 6.7.** The proof of (iii) is omitted because it is similar to (ii). Here is an informal presentation of the inductive argument common to both. Define

where

The rows and columns of are labelled by the divisors of , as indicated below for the case :

Say that a partition of is *coprime* if the highest common factor of the numbers in each part is . The aim of the game is to find linear combinations of the rows of , always including rows and , such that the subsets of the columns on which the row sums are equal form a coprime partition of the divisors.