PERMUTATION GROUPS CONTAINING A REGULAR
ABELIAN SUBGROUP: THE TANGLED HISTORY OF
TWO MISTAKES OF BURNSIDE

arXiv:1705.07502v2 [] 11 Jun 2017

MARK WILDON

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 2transitive. 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.

1. Introduction
In 1911, writing in §252 of the second edition of his influential textbook [6],
Burnside claimed a proof of the following theorem.
Theorem 1.1. Let G be a transitive permutation group of composite primepower degree containing a regular cyclic subgroup. Either G is imprimitive
or G is 2-transitive.
An error in the penultimate sentence of Burnside’s proof was noted in
[7, page 24], where Neumann remarks ‘Nevertheless, the theorem is certainly
true and can be proved by similar character-theoretic methods to those
that Burnside employed’. In §3 we present the correct part of Burnside’s
proof in today’s language. In §4 we prove Theorem 1.1 using the lemma
on cyclotomic integers in §2 below. In §5 we build on the correct part of
Burnside’s proof in a different way, obtaining an entirely character-theoretic
proof of the following variation on Theorem 1.1.
Theorem 1.2. Let G be a transitive permutation group of composite nonprime-power degree containing a regular cyclic subgroup. Either G is imprimitive or G is 2-transitive.
Date: June 13, 2017.
2010 Mathematics Subject Classification. 20B05, secondary 20B15, 20C20.

2

MARK WILDON

In honour of Burnside, Wielandt [37, §25] defined a B-group to be a
group K such that every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. Thus Theorems 1.1 and 1.2 imply
that cyclic groups of composite order are B-groups.
The early attempts to prove this result by character-theoretic methods
are rich with interest, but also ripe with errors. Our second aim, which
occupies §6, is to untangle this mess. We end in §7 with some new results on
abelian B-groups which require the Classification Theorem of Finite Simple
Groups. We state an open problem on when C2n is a B-group, present a
partial solution, consider B-groups of prime-power order and make some
further (much more minor) corrections to the literature.
At a late stage in this work, the author learned of [25], in which Knapp
gives another way to fix Burnside’s proof of Theorem 1.1, using essentially
the same lemma as in §2. The key step in Knapp’s proof is his Proposition 3.1. It uses two compatible actions of the Galois group of Q(ζ) : Q,
where ζ is a root of unity of order the degree of G: firstly on the set permuted by G, and secondly on the corresponding permutation module. The
proof of Theorem 1.1 given here uses only the second action (in a simple
way that is isolated in the second step), and is more elementary in several
other respects. The inductive approach in our third step is also new. Given
the historical importance of Theorem 1.1, the author believes it is worth
putting this shorter proof on record. Theorem 1.2 is not proved in [25].
2. Lemma on cyclotomic integers
The following lemma is essentially the same as Lemma 4.1 in [25]. A
proof is included for completeness. Recall that the degree of the extension
of Q generated by a primitive d-th root of unity is φ(d), where φ is Euler’s
totient function.
Lemma 2.1. Let p be a prime and let n ∈ N. For each r such that 1 ≤ r <
pn−1 , let
R(r) = {r, r + pn−1 , . . . , r + (p − 1)pn−1 }.
Ppn −1
n−1
ai ζ i ∈
Let ζ be a primitive pn -th root of unity and let ω = ζ p . If i=0
Q[ω] where ai ∈ Q for each i, then the coefficients ai are constant for i in
each set R(r).
Proof. By the Tower Law [Q(ζ) : Q(ω)] = [Q(ζ) : Q]/[Q(ω) : Q] =
n−1
− ω is
φ(pn )/φ(p) = (p − 1)pn−1 /(p − 1) = pn−1 . Therefore Ψ(X) = X p
the minimal polynomial of ζ over Q(ω). By hypothesis there exists γ ∈ Q[ω]
such that
X
f (X) = −γ +
ai X i
0≤i<pn

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

3

has ζ as a root. Hence f (X) is divisible in Q(ω)[X] by Ψ(X). There is a
P
unique expression f (X) = f0 (X) + 0<r<pn−1 fr (X) where
X
fr (X) =
ai X i
0≤i<pn
i≡r mod pn−1

for 0 < r < pn−1 . The remainder when X d is divided by Ψ(X) has non-zero
coefficients only for those X c such that c is congruent to d modulo pn−1 .
Therefore each fr (X) is divisible by Ψ(X) and so fr (ζ) = 0 for each r.
Since the coefficients of fr for 0 < r < pn−1 are rational, it follows that each
such fr is divisible, now in Q[X], by the minimal polynomial of ζ over Q,
n−1
n−1
+ · · · + X (p−1)p . Since fr has degree at most
namely Φpn (X) = 1 + X p
pn − 1, this implies that fr (X) = br X r Φpn (X) for some br ∈ Q. The lemma
follows.

3. Burnside’s method: preliminary results
We may suppose that G acts on {0, 1, . . . , d−1}, where d ∈ N is composite,
and that g = (0, 1, . . . , d − 1) is a d-cycle in G. Let H be the point stabiliser
of 0. Let M = he0 , e1 . . . , ed−1 iC be the natural permutation module for G.
Let ζ be a primitive d-th root of unity and for 0 ≤ j < d let
X
(1)
vj =
ζ −ij ei .
0≤i<d

Since ei g = ei+1 , where subscripts are taken modulo d, we have vj g = ζ j vj
P
for each j. Note that v0 = 0≤i<d ei spans the (unique) trivial CG-module
of M . Let
(2)

M = hv0 i ⊕ V1 ⊕ · · · ⊕ Vt

be a direct sum decomposition of M into irreducible CG-submodules. The vj
are eigenvectors of g with distinct eigenvalues. Therefore they form a basis
of M . Moreover, since the eigenvalues are distinct, each of the summands
V1 , . . . , Vt has a basis consisting of some of the vj . Thus the decomposition
in (2) is unique. For each summand Vk , let Bk = {j : 0 < j < pn , vj ∈ Vk }.
Let φk be the character of Vk .
The following two lemmas are the key observations in Burnside’s method.
Lemma 3.1. For each k such that 1 ≤ k ≤ t, the vector
invariant.

P

j∈Bk

vj is H-

P
Proof. The permutation character π of G is 1G + tk=1 φk , where the summands are distinct and irreducible. By Frobenius reciprocity we have
xG

1 = hπ, φk iG = h1H  , φk iG = h1H , φk yH iH

4

MARK WILDON

for each k. Therefore each Vk has a unique 1-dimensional CH-invariant
P
submodule. Since e0 = p1d 0≤j<d vj is H-invariant, and the projection
P
P
of e0 into Vk is p1d j∈Bk vj , this submodule is spanned by j∈Bk vj .


Lemma 3.2. If O is an orbit of H on {0, 1, . . . , d − 1} and 1 ≤ k ≤ t then
P
the sum i∈O ζ ij is constant for j ∈ Bk .
P
Proof. Observe that k∈O ek is H-invariant. An easy calculation (which
may be replaced by the observation that the character table of Cd is an
P
orthogonal matrix) shows that ei = p1d 0≤j<d ζ ij vj for each i. Therefore
X
X X 
ei =
ζ ij vj .
i∈O

0≤j<d i∈O

By Lemma 3.1 the coefficients are constant for j ∈ Bk .



The following proposition is used in the final step of the proof of both
main theorems.
Proposition 3.3. If there is a prime p dividing d and a summand Vk whose
basis {vj : j ∈ Bk } contains only basis vectors vj with j divisible by p then
there exists a normal subgroup of G containing g d/p whose orbits form a
non-trivial block system.
Proof. Let N be the kernel of G acting on Vk . Since vj g = ζ j vj , N conP
tains gd/p . By Lemma 3.1, Vk has h j∈Bk vj i as an HN -invariant subspace.
Since Vk is not the trivial module, we have HN < G. Hence N is nontrivial but intransitive. The orbits of the normal subgroup N are blocks of
imprimitivity for G.

4. Proof of Theorem 1.1
First step. By hypothesis G has degree pn where p is prime and n ≥ 2. The
Galois group Gal(Q(ζ) : Q) of the field extension Q(ζ) : Q permutes the
basis vectors vj while preserving the unique direct sum decomposition (2).
Hence Gal(Q(ζ) : Q) permutes the sets B1 , . . . , Bt . By Proposition 3.3, we
may assume that every Bk contains some j not divisible by p. Hence, given
any m such that 0 < m < n, there exists j not divisible by p such that the
set Bk containing pm also contains j. Let Bℓ be the set containing 1. Since
the Galois group is transitive on {ζ j : 0 < j < pn , p ∤ j}, by conjugating ζ j
to ζ, we see that pm c ∈ Bℓ for some c not divisible by p.
Let P be the partition of {1, . . . , pn − 1} into the orbits of H other
than {0}. The previous paragraph and Lemma 3.2 imply that for all m
such that 0 < m < n there exists cm ∈ N, not divisible by p, such that
X
X m
ζi =
ζ p cm i
(3)
i∈O

for each O ∈ P.

i∈O

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

5

Second step. We shall show by induction on n that (3) implies that P is
the one-part partition. It then follows that H is transitive on {1, . . . , pn − 1}
and so G is 2-transitive, as required.
Fix O ∈ P. Taking m = n − 1 in (3) and applying Lemma 2.1 with
P
n−1
ω = ζ p cn−1 , we find that the coefficients in i∈O ζ i are constant on the
sets R(r) = {r, r + pn−1 , . . . , r + (p − 1)pn−1 } for 0 < r < pn−1 . Hence O is a
union of some of these sets, together with some of {pn−1 }, . . . , {(p−1)pn−1 }.
The contributions from R(r) to (3) are
(4)
X

(5)

X

ζ i = 0,

i∈R(r)

ζp

mc

mi

= pζ p

mc

mr

.

i∈R(r)

Case n = 2. Let ω = ζ pc1 . Taking m = 1 in (3) and substituting the
relations in (4) and (5) we get
X

r∈O
0<r<p

0+

X

X

ωi =

pi∈O

X

pω r +

r∈O
0<r<p

1.

pi∈O
0<i<p

This rearranges to
{O ∩ {p, 2p, . . . , (p − 1)p} +

X

(p[i ∈ O] − [pi ∈ O])ω i = 0,

0<i<p

where the Iverson bracket [P ] is 1 if the statement P is true, and 0 if false.
Since the minimal polynomial of ω, namely 1+X+· · ·+X p−1 , has degree p−1
and constant coefficients, it follows that {O ∩ {p, . . . , (p − 1)p} = p − 1 and
i ∈ O for each i such that 0 < i < p. Thus O = {1, . . . , p2 − 1} as required.
Inductive step. Let n ≥ 3. Let T = {pn−1 , . . . , (p−1)pn−1 }. Substituting (5)
in the right-hand-side of (3) for first m = 1 and then a general m such that
0 < m < n, we have
X

pζ pc1r + |O ∩ T | =

r∈O
0<r<pn−1

X

pζ p

mc

mr

+ |O ∩ T |.

r∈O
0<r<pn−1

For each O ∈ P, define O⋆ = O ∩ {1, . . . , pn−1 − 1}. Clearly {O⋆ : O ∈ P} is
a set partition of {1, . . . , pn−1 − 1}. Let ζ⋆ = ζ pc1 and, for each m such that
0 < m < n, choose dm ∈ N such that c1 dm ≡ cm mod p. We may suppose
that d1 = 1. Replacing r with i⋆ , the previous displayed equation implies
X

i⋆ ∈O⋆

ζ⋆i =

X

i⋆ ∈O⋆

ζ⋆p

m−1 d i
m ⋆

.

6

MARK WILDON

Comparing with (3), we see that all the conditions are met to apply the
inductive hypothesis. Hence O⋆ = {1, . . . , pn−1 − 1} and so O contains
P
{1, . . . , pn − 1}\T . By (4) and (5) we have i∈{1,...,pn−1}\T ζ i = 0 and
X

ζ pc1 i = p

0<i<pn−1
i6∈T

X

ζ⋆i = −p.

0<i<pn−1

Substituting these two results in the case m = 1 of (3) we get
X
n−1
ζ p i = −p + |O ∩ T |.
pn−1 i∈O∩ T

It follows, as in the final step of the case n = 2, that |O ∩ T | = p − 1 and so
O ⊇ T and O = {1, . . . , pn − 1}, as required.
5. Proof of Theorem 1.2
We continue from the end of §3. Let ϑ : hgi → C be the faithful linear
character of hgi defined by ϑ(g) = ζ. For 1 ≤ k ≤ t, let πk be the character
P
of Vk restricted to hgi. Since hvj i affords ϑj , we have πk = j∈Bk ϑj . Since
the sets B1 , . . . , Bt are disjoint, the characters πk are linearly independent.
Let p be a prime dividing d. The character of Vk⊗p is πkp . Since (a + b)p ≡
ap + bp mod p for all a, b ∈ Z, we have
X
{j ∈ Bk : jp ≡ rp mod d} ϑrp + pπ
(6)
πkp =
0≤r<d/p

P
for some character π of hgi. Let πkp − pπ = a1H + tℓ=1 aℓ πℓ . By the linear
independence of the πℓ , it follows from (6) that if aℓ 6= 0 then πℓ contains
only characters of the form ϑrp with 1 ≤ r < d/p. Thus for any such ℓ, Bℓ
contains only basis vectors vj with j divisible by p and, by Proposition 3.3, G
is imprimitive. We may therefore assume that {j ∈ Bk : jp ≡ rp mod d} is
a multiple of p for each r such that 1 ≤ r < d/p. Identifying {0, 1, . . . , d − 1}
with Z/dZ, note that jp ≡ rp mod d if and only if j ∈ r + hd/pi. Therefore
for each prime p dividing d, each Bk is the union of a subset of hd/pi and
some proper cosets r + hd/pi.
Let q be a prime dividing d other than p. Since the subgroups hd/pi and
hd/qi of Z/dZ meet in 0, each member of hd/pi\{0} is in a proper coset
of hd/qi, and similarly with p and q swapped. By the conclusion of the
previous paragraph, if Bk meets hd/pqi then Bk contains hd/pqi\{0}. At
most one Bk has this property. If t = 1 then G is 2-transitive, so we may
assume that d > pq and there exists Bk not meeting hd/pqi. For this Bk
there exist r1 , . . . , rs such that 0 < r1 < . . . < rs < d/pq and
Bk =

s
[

(re + hd/pqi).

e=1

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

7

Thus |Bk | = spq and
(7)

πk πk = s(ϑ0 + ϑd/pq + · · · + ϑ(pq−1)d/pq ) + ψ

where the coefficient of ϑj in ψ is equal to the number of pairs (e, e′ ) such
that j ∈ −re + re′ + hd/pqi. There are exactly s such pairs if and only if for
all e there exists a unique e′ such that re + j + hd/pqi = re′ + hd/pqi, or,
equivalently, if and only if Bk + j = Bk , where the addition is performed in
Z/dZ. Let
J = {j ∈ Z/dZ : Bk + j = Bk }.
Since J is a subgroup of Z/dZ containing d/pq we have J = hmi for some m
dividing d/pq. Since 0 6∈ Bk , and so −r1 , . . . , −rs 6∈ J, we have m > 1.
Thus (7) may be rewritten as

π k πk = s ϑ0 + ϑm + · · · + ϑn−m + φ

where hφ, ϑj i < s for all j not divisible by m. By the linear independence of
π1 , . . . , πt , there exists πk such that if hπk , ϑj i > 0 then j is a multiple of m.
The result now follows from Proposition 3.3.
6. A historical survey of Burnside’s method and B-groups
6.1. Burnside’s work for prime-power degree. We begin in 1901 with
[3, §7], in which Burnside used character-theoretic arguments to prove the
following important dichotomy. (All of the papers of Burnside discussed
below appear in Volume II of his collected works [8].)
Theorem 6.1 (Burnside 1901 [3, §7]). A permutation group of prime degree p is either 2-transitive or contains a normal subgroup of order p.

In the following §8 he proves Theorem 1.1 for permutation groups of odd
degree p2 using character theory. He comments ‘It appears highly probable
that this result may be extended to any group of odd order which contains
a regular substitution of order equal to the degree of the group; but I have
not yet succeeded in proving this.’
In the revised second edition of his textbook [6], Burnside added five entirely new chapters on linear groups and characters. Most notably these include the well-known character-theoretic proof of the pa q b -Theorem. In §251
he used the method of cyclotomic sums and basis sets, introduced in his
1906 paper [4, §7] but presented in his textbook with some simplifications,
to prove Theorem 6.1. The following §252, whose correct part was presented
in §3, attempts to prove Theorem 1.1. Burnside’s argument appears to have
been generally accepted, both at the time and later, until Neumann pointed
out the error in his essay in [38]. For example, it is cited without critical
comment by Wielandt in [37]. Its mistake is to assert that the only solutions
to (3) when m = n − 1 have |O| = pn − 1. This gives one solution, but there
are others.

8

MARK WILDON

Recall that if 1 ≤ r < pn−1 then R(r) = {r, r + pn−1 , . . . , r + (p − 1)pn−1 }.
Define Z ⊆ {1, . . . , pn − 1} to be null if there exists s ∈ N0 and distinct
rij ∈ {1, . . . , pn−1 − 1} for 0 ≤ i ≤ p − 1 and 1 ≤ j ≤ s such that rij ≡ i
S
Ss
mod p for each i and j and Z = p−1
i=0
j=1 R(rij ).
Proposition 6.2. Let n ≥ 2 and let ω be a primitive p-th root of unity. Let
O ⊆ {1, . . . , pn − 1}. Then
X
X
ζi =
ωi
i∈O

i∈O

if and only if either
(i) O is null; or
S
(ii) O = {pn−1 , . . . , (p − 1)pn−1 } ∪ p−1
i=1 R(ri ) ∪ Z where Z is null, the
ri are distinct elements of {1, . . . , pn−1 − 1}\Z and ri ≡ i mod p for each i.
The proof is similar to the inductive step in §4, using (4) and (5) to show
P
P
i
i
that if Z is null then
i∈Z ξ =
i∈Z ω = 0, and Lemma 2.1 to show
that O\{pn−1 , . . . , (p − 1)pn−1 } is a union of the sets R(r). Note that since
r01 ≡ 0 mod p, and 1 ≤ r01 < pn−1 , Case (i) is relevant only when n ≥ 3.
The smallest possible O has size p2 − 1, coming from Case (ii); this shows
Burnside’s claim is false whenever p ≥ 3 or n ≥ 3. The lack of structure in
the solutions, beyond that captured by the sets R(r), suggests that any fix
to Burnside’s proof must involve significant further ideas.
6.2. Burnside’s 1921 paper. In [5], Burnside claimed a ‘remarkably simple’ proof that every abelian group that is not elementary abelian is a Bgroup, as conjectured at the end of §252 of [6]. (Of course Burnside did not
use the term ‘B-group’.) The groups Sd ≀ S2 in their primitive action for d
composite, seen in Example 6.3 below, show that this result is false. In [31,
§15], D. Manning raised this family of counterexamples and observed ‘the
first and most important part of the proof must contain a serious mistake’.
In today’s language, Burnside considers a permutation group G of degree
′
dd acting on {0, . . . , d − 1} × {0, . . . , d′ − 1}, containing a regular subgroup
K = hgd i × hgd′ ′ i where gd = (0, 1, . . . , d − 1) and gd′ ′ = (0, 1, . . . , d′ − 1).
The natural CG-permutation module M factorizes on restriction to K as
he0 , . . . , ed−1 i ⊗ he′0 , . . . , e′d′ −1 i. Let ζd , ζd′ ∈ C be primitive roots of unity of
orders d and d′ , respectively. The analogue of the vj basis element defined
earlier in (1) is
X −ij
X −i′ j ′
v(j,j ′ ) =
ζd ei ⊗
ζd′ e′i′
0≤i<d

0≤i′ <d′

where 0 ≤ j < d and 0 ≤ j ′ < d′ . As before, M has a unique decomposition
hv(0,0) i⊕V1 ⊕· · ·⊕Vt where each irreducible summand Vk has a basis {v(j,j ′ ) :
(j, j ′ ) ∈ Bk } for some subset Bk of {0, . . . , d − 1} × {0, . . . , d′ − 1}. Let φk

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

9

be the character of Vk . The analogue of Lemma 3.2 is that if O is an orbit
of the point stabiliser H of (0, 0), and 1 ≤ k ≤ t then
X ij i′ j ′
(8)
ζd ζd′
(i,i′ )∈O

is constant for (j, j ′ ) ∈ Bk . Burnside proves this, and also proves (in a similar
P
′ ′
′
way) the dual relation that the character value φk (gdi gd′i′ ) = (j,j ′ )∈Bk ζdij ζdi ′j
is constant for (i, i′ ) ∈ O. Hence
X
X ij i′ j ′
X
′ ′
(9)
ζd ζd′
ζdij ζdi ′j = |Bk |
(i,i′ )∈O (j,j ′ )∈Bk

(i,i′ )∈O

= |O|

(10)

X

′ ′

ζdij ζdi ′j

(j,j ′ )∈Bk

provided (j, j ′ ) ∈ Bk in the right-hand side of (9) and (i, i′ ) ∈ O in the
right-hand side of (10). Burnside chooses Bk to contain (d/q, 0) where q is a
prime factor of d and O to contain (1, 0). By taking (j, j ′ ) = (d/q, 0) in (9)
P
P
id/q
and (i, i′ ) = (1, 0) in (10) he obtains |Bk | (i,i′ )∈O ζd = |O| (j,j ′ )∈Bk ζdj =
|O|φk (gd ), and so
|Bk | X i
ω
(11)
φk (gd ) =
|O|
′
(i,i )∈O

d/q

where ω = ζd is a primitive root of unity of order q.
The fourth displayed equation on page 484 of [5] claims that φk (gdq ) = |Bk |,
and so gdq is in the kernel of φk . It appears that Burnside substitutes gdq
for gd in (11), and replaces ω with ω q . If (11) expressed φk (gd ) as a sum
of eigenvalues, as in (10), this would be legitimate. However this is not the
case, and the following example shows that Burnside’s claim is in general
false.
Example 6.3. Let d ∈ N. Let S be the symmetric group on the set
{0, 1, . . . , d − 1}. Let N = S × S and let G ∼
= S ≀ C2 be the wreath product
N ⋊hτ i where τ has order 2 and acts on N by (g, g ′ )τ = (g ′ , g). In the action
of G on {0, 1, . . . , d − 1}2 , the point stabiliser H of (0, 0), namely (T × T ) ⋊ τ
where T is the symmetric group on {1, . . . , d − 1}, has two non-singleton


orbits: (j, 0), (0, j) : 1 ≤ j < d and (j, j ′ ) : 1 ≤ j, j ′ < d . Therefore G
is not 2-transitive. Provided d ≥ 3, H is a maximal subgroup of G, so G
is primitive. Let gd = gd′ = (0, 1, . . . , d − 1). Since hgd i × hgd′ i ≤ N acts
regularly, Cd × Cd is not a B-group whenever d ≥ 3.
Let d ≥ 3. The natural permutation character of S is 1S + χ where χ is
irreducible. By the branching rule (see [21, Ch. 9] or [20, Lemma 2.3.10]),
χ is the unique non-trivial character of S whose restriction to T contains
the trivial character. By the classification of irreducible characters of wreath
products [20, Theorem 4.3.34], it follows that the irreducible characters of G

10

MARK WILDON
f

that contain the trivial character on restriction to H are 1G , φ and χ×2 ,
f
G and χ×2
where φ = (χ×1S )↑N
is the unique character of G whose restriction
to N is χ × χ. By Frobenius reciprocity, the permutation character of
f
M is 1G + φ + χ×2 . Considering restrictions to hgd i × hgd′ i, we get M =
hv(0,0) i ⊕ hv(j,0) , v(0,j ′ ) : 1 ≤ j < d, 1 ≤ j ′ < di ⊕ hv(j,j ′ ) : 1 ≤ j, j ′ < di.
The second summand has character φ and contains v(1,0) and v(0,1) , so is
a faithful CG-module. Thus, contrary to Burnside’s claim, no non-identity
power of gd is in the kernel of φ. Burnside’s conclusion, that G has a proper
normal subgroup containing gdq holds, since we may take the base group N ,
but clearly Burnside intends the normal subgroup to be the kernel of φ, so
that Proposition 3.3 can be applied, and the kernel of φ is trivial.
The penultimate paragraph of Burnside’s paper considers the case where d
and d′ are distinct primes. This is the hardest part of the paper to interpret:
the claims are correct, but the argument has a significant gap. Burnside has
already assumed that G is not 2-transitive. If a basis set Bk is contained in
{(1, 0), . . . , (d − 1, 0)} then, identifying (j, j ′ ) with d′j + dj ′ mod dd′ , Proposition 3.3 implies that G has a normal intransitive subgroup N containing
hgd i. This gives the first of Burnside’s claims. While not stated explicitly,
it seems that Burnside then assumes, as he may, that no Bk is contained
in {(1, 0), . . . , (d − 1, 0)}. He makes two further claims, equivalent to the
following:
(i) If Bk meets {(1, 0), . . . , (d − 1, 0)} then Bk is a union of sets of the

form (j, 0), (j, 1), . . . , (j, d′ − 1) where 1 ≤ j < d.
(ii) there is a set Bℓ contained in {(0, 1), . . . , (0, d′ − 1)}.
Clearly (i) implies (ii), and by Proposition 3.3, (ii) implies that G has
a normal intransitive subgroup N containing hgd′ ′ i. To prove (i), we use
the italicised conclusion of the second paragraph in the proof of Theorem 1.2 in §5: taking p = d′ , this implies that Bk is the union of a subset of {(0, 1), . . . , (0, d′ − 1)} and some sets of the required form. Since
[Q(ζdd′ ) : Q(ζd )] = φ(dd′ )/φ(d) = φ(d′ ) = [Q(ζd′ ) : Q], the stabiliser
of ζd in the Galois group Gal(Q(ζdd′ ) : Q) acts transitively on the roots
′
ζd′ , . . . , ζdd′ −1 . By the hypothesis in (i) there exists (j, 0) ∈ Bk . For each r ′
′
such that 1 ≤ r ′ < d′ there exists σ ′ ∈ Gal(Q(ζdd′ ) : Q) such that ζdσ = ζd
′
′
σ′
σ′
and ζdσ′ = ζdr′ . Since v(j,0)
= v(j,0) and v(0,1)
= v(0,r′ ) , we see that if Bk
meets {(0, 1), . . . , (0, d′ − 1)} then it contains this set; a similar argument,
taking σ ∈ Gal(Q(ζdd′ ) : Q) such that ζdσ = ζdr and ζdσ′ = ζd′ now shows
that Bk = {0, . . . , d − 1} × {0, . . . , d′ − 1}\{(0, 0)}, and so G is 2-transitive,
contrary to assumption. Therefore (i) holds.
Having proved (i), we instead follow Burnside’s argument for (i) and (ii).
Burnside chooses O to contain (1, 1) and takes (m, 0) ∈ Bk . By (9) and (10),
P
P
P ′ −1
′
|O| jd′ =0
cj ′ ζdj′ = |Bk | (i,i′ )∈O ζdim , where cj ′ = j:(j,j ′)∈Bk ζdj for j ′ ∈

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

11

{0, 1, . . . , d′ −1}. According to Burnside, this implies that the coefficients cj ′
are constant for all j ′ . It appears that Burnside assumes that every rational
′
relation between the powers of ζd′ is a multiple of 1 + ζd′ + · · · + ζdd′ −1 . But a
′
more general relation is a + ζd′ + · · · + ζdd′ −1 = a − 1, so we can only conclude
that the cj ′ are constant for j ′ ∈ {1, . . . , d′ − 1}. However, it is true that if
P
P
j
j
j∈J ζd =
j∈K ζd for non-empty sets J, K ⊆ {0, 1, . . . , d−1} then J = K,
so this weaker conclusion implies that, for each j ′ ∈ {1, . . . , d′ − 1}, either
{j : (j, j ′ ) ∈ Bk } ⊇ {1, . . . , d − 1} or {j : (j, j ′ ) ∈ Bk } ⊆ {0}. Hence
(i)′ If Bk meets {(1, 0), . . . , (d − 1, 0)} then Bk is a union of sets of the
form {(j, 0)} and {(j, 1), . . . , (j, d′ − 1)} where 1 ≤ j < d.
The Galois action of the automorphisms σ in our proof of (i) shows that (i)′
implies (ii). Therefore Burnside’s argument can be corrected.
The final sentence of the paragraph we have been reading is ‘It is clear that
the same method of proof will apply, when the transitive Abelian subgroup
has three or more independent generators’. Taking d = 4 in Example 6.3,
we see that the subgroup h(0, 1, 2, 3)i × h(0, 1)(2, 3), (0, 2)(1, 3)i ≤ G acts
regularly in the primitive action of G on {0, 1, 2, 3}2 . Therefore C4 × C2 × C2
is not a B-group and Burnside’s claim is false. The use of the Galois action
in the previous paragraph required that both d and d′ are prime.
In §6.5 below we extend the correct part of Burnside’s proof to show that
if p is an odd prime and n ∈ N then C2n , C2n p and C2pn are B-groups. A
proof of Conjecture 6.5 will rehabilitate Burnside’s method for cyclic groups.
6.3. Manning’s 1936 paper. In [31], D. Manning claimed a proof, using
Burnside’s method, that if p is prime and a > b then Cpa × Cpb is a B-group.
It is reported in [37, page 67] that she later acknowledged that the critical
Lemma II in [31] is false. We extend Example 6.3 to show this.
Example 6.4. Recall from Example 6.3 that S is the symmetric group on
{0, 1, . . . , d − 1} and G ∼
= S ≀ C2 acting primitively on {0, 1, . . . , d − 1}2 .
We took gd = gd′ = (0, 1, . . . , d − 1). By Example 6.3, the natural CGpermutation module has a summand with basis set B = {(j, 0), (0, j ′ ) : 1 ≤
j < d, 1 ≤ j ′ < d}, with respect to the chosen generators (gd , 1) and (1, gd′ )
of the regular subgroup K = hgd i × hgd′ i.
We have
v(j,0) (gd , 1) = ζ j v(j,0) ,
v(j,0) (gd , gd′ ) = ζ j v(j,0) ,

v(0,j ′ ) (gd , 1) = v(0,j ′ ) ,
′

v(0,j ′ ) (gd , gd′ ) = ζ j v(0,j ′ ) .

Therefore, with respect to the alternative generators (gd , 1) and (gd , gd′ )
of K, the basis set becomes C = {(j, j) : 1 ≤ j < d} ∪ {(0, j ′ ) : 1 ≤ j ′ < d}.
Observe that, as it must be, C is invariant under the action induced by
Gal(Q(ζd ) : Q). Manning’s Lemma II asserts the stronger property that,

12

MARK WILDON

given any (i, i′ ) ∈ {0, 1, . . . , d − 1}2 with i and i′ coprime to d, C is invariant under the permutation (j, j ′ ) 7→ (ij, i′ j ′ ), where the entries are taken
modulo d. Taking i = 1 and i′ = −1 we see that this is false whenever
d > 2.
6.4. Later proofs of Burnside’s and Manning’s claims. In 1908, Schur
introduced his method of S-rings and gave the first correct proof of Theorem 1 [34]. In 1933 Schur extended his method to prove, more generally,
that any cyclic group of composite order is a B-group. As remarked in
[31], it appears that Schur was unaware of Burnside’s 1921 paper. In 1935,
Wielandt wrote ‘Der von Herrn Schur angegebene Beweis ist recht schwerig’, and gave a short proof of the still more general result that any abelian
group of composite order having a cyclic Sylow p-subgroup for some prime p
is a B-group [36]. Wielandt’s proof depends on several results on S-rings, in
particular property (6) in [36], that the stabiliser of an element of an S-ring
is itself in the ring. Wielandt’s result and proof appear, in translation but
essentially unchanged, in his 1964 textbook [37, Theorem 25.4]. The use of
complex conjugation at the end of the proof of Theorem 1.2 in §5 involves
some similar ideas to the proof of property (6) in [37, Theorem 23.5], but
the proof here is substantially shorter and more elementary.
The first essentially correct proof of the result claimed by D. Manning
was given by Kochendörffer in 1937 using S-rings [26]; Wielandt comments
in [37] that it is ‘very complicated’ (Bercov’s translation). In his essay in [38],
Neumann reports that in an unpublished note D. Manning found some slips
in [26], but was able to correct them. Neumann’s essay includes a proof
of Theorem 1.1 that a reader, familiar with the prerequisites from modular
representation theory and permutation groups, will find spectacularly short
and beautiful.
Apart from [25], outlined in the introduction, the three papers [3, 5, 31]
surveyed in this section appear to exhaust the research literature on Burnside’s method. It is intriguing that all err in ultimately the same way, by
overlooking algebraic relations satisfied by roots of unity.
6.5. Burnside’s method in even degree. Again we continue from the
end of §3. There is an action of the Galois group Gal(Q(ζd ) : Q) on the
set {1, . . . , d − 1} under which σ ∈ Gal(Q(ζd ) : Q) sends i to i′ if and only
′
if σ sends ζ i to ζ i . In [25, Theorem 2.3(2)], Knapp extends Burnside’s
arguments to show that this action induces an action of the Galois group on
the orbits of the point stabiliser H. (This result may also be proved using
S-rings: see [37, Theorem 23.9].) Let D be the set of divisors of d. Set
O(1) = {0} and for r ∈ D with r > 1, set
O(r) = {md/r : 0 < m < r, hcf(m, r) = 1}.

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

13

Thus for each r ∈ D the set {ζdi : i ∈ O(r)}, consisting of all primitive
r-th roots of unity, is an orbit of the Galois group on the powers of ζd .
d/2
If d is even then, since O(2) = {d/2} corresponds to ζd = −1 ∈ Q,
the H-orbit O containing d/2 is invariant under the Galois action. Hence
S
O = r∈E O(r) for some subset E of D. Observe that G is 2-transitive if
and only if E = D\{1}.
P
P
For r ∈ D and j ∈ N we have i∈O(r) ζdij = α αj where α ranges over
all primitive r-th roots of unity. If hcf(r, j) = j ⋆ then the map α 7→ αj is j ⋆
to 1, and each αj is a primitive r/j ⋆ -th root of unity. It is well known that
the sum of the φ(s) roots of unity of order s is µ(s), where µ is the Möbius
function (see for instance [35, Exercise 2.8]). Therefore, if R is the matrix
with rows and columns indexed by D, defined by
 φ(r)

r
(12)
Rrc = µ
r
hcf(r, c) φ( hcf(r,c)
)
then, for any r ∈ D and j ∈ N,
X ij
(13)
ζd = Rrc

where c = hcf(d, j).

i∈O(r)

(Here R stands for Ramanujan, who considered these cyclotomic sums in [33];
this was published in the interval between Burnside’s 1901 and 1921 papers,
but there is no evidence that Burnside was aware of its relevance.) As an
aide-memoire, we note that Rrc is defined by taking c-th powers of r-th
roots of unity. An example of these matrices is given after Lemma 6.6.
Let ∼E be the relation on D\{d} defined by
X
X
(14)
b ∼E c ⇐⇒
Rrb =
Rrc .
r∈E

r∈E

Let PE be the set of equivalence classes of ∼E . Given B ⊆ {0, 1, . . . , d − 1},
let Y (B) = {c ∈ D\{d} : B ∩ O(d/c) 6= ∅}. For example, 1 ∈ Y (B) if and
only if B contains a number coprime to d, and Y ({0}) = ∅. If Bk and Bℓ
are distinct basis sets then necessarily Bk ∩ Bℓ = ∅, but if neither Bk nor Bℓ
is invariant under the Galois action, we may still have Y (Bk ) ∩ Y (Bℓ ) 6= ∅.
However the asymmetry between orbits and basis sets in the conclusion
P
of Lemma 3.2 works in our favour, to show that r∈E Rrc is constant for
c ∈ Y (Bk ). It follows that Y (Bk ) is contained in a single part of the partition
PE of D\{d}. Hence, by Proposition 3.3, we may assume that the highest
common factor of the entries in each part of the partition PE of D\{d} is 1.
We say that such partitions are coprime.
For c ∈ D, an easy calculation from (13) shows that
(
d/c−1
d−1
X
X
X
0 if c < d,
i
ic
ζd/c =
ζd = c
Rrc =
d if c = d.
i=0
i=0
r∈D

14

MARK WILDON

Since R1c = 1 for all c ∈ D, it follows that if E = D\{1} then PE =

D\{d} . This proves the ‘only if’ direction of the following conjecture.
Conjecture 6.5. Let E ⊆ D contain 2. The partition PE of D\{d} defined
by the relation ∼E in (14) is coprime if and only if E = D\{1} or E = D.

We have shown that if d is even then, defining E as above by the orbit O
containing d/2, the ‘if’ direction of Conjecture 6.5 implies that E = D\{1}
and O = {1, . . . , d − 1}, and so Cd is a B-group.
The following lemma can be used to prove Conjecture 6.5 in several cases
of interest. Let R(d) denote the Ramanujan matrix defined for degree d.
Lemma 6.6.
(i) Let p be prime and let n ∈ N. We have


if f < e − 1,

0
R(pn )pe pf = −pe−1
if f = e − 1,


(p − 1)pe−1 if f ≥ e.

(ii) Let p1 , . . . , ps be distinct primes and let n1 , . . . , ns ∈ N. We have
R(d) = R(pn1 1 ) ⊗ · · · ⊗ R(pns s ).

Proof. Part (i) is immediate from (12). For (ii), it suffices to show that if d
and d′ are coprime and r | d, r ′ | d′ and c | d, c′ | d′ then the entry in row
rr ′ and column cc′ of R(dd′ ) is Rrc (d)Rr′ c′ (d′ ). This follows from (12) using
the multiplicativity of µ and φ, noting that hcf(r, r ′ ) = hcf(c, c′ ) = 1.

For example, if p is an odd prime then R(2p3 ) is as shown below, with D
ordered 1, 3, 9, 27, 2, 6, 18, 54 and row 2 ∈ E highlighted. The division indicates the tensor factorization R(p3 ) ⊗ R(2).

1
1
1
1
1
1
1
1
 −1
p−1
p−1
p−1
−1 p − 1
p−1
p−1 




 0
−p
p(p − 1)
p(p − 1)
0
−p p(p − 1) p(p − 1) 


 0
0
−p2
p2 (p − 1)
0
0
−p2
p2 (p − 1)



−1
−1
−1
−1
1
1
1
1




−(p − 1) −1 p − 1
p−1
p−1 
 1 −(p − 1) −(p − 1)


 0
p
−p(p − 1) −p(p − 1)
0
−p p(p − 1) p(p − 1) 
0
0
p2
−p2 (p − 1) 0
0
−p2
p2 (p − 1)


In particular R(p3 ) appears as the top-left block.

Proposition 6.7. Let n ∈ N and let p be an odd prime. Conjecture 6.5
holds when (i) d = 2n , (ii) d = 2n p and (iii) d = 2pn .
Proof. The ‘if’ direction remains to be proved. Recall that the rows and
columns of R are labelled by the divisors of d. Since row 1 of R(d) is
constant, we may assume that 1 ∈ E.
Suppose, as in (i), that d = 2n . If n = 1 then E = {1, 2} and the conclusion is immediate. Suppose that n ≥ 2. Let R⋆ be the matrix obtained

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

15

from R(2n ) by deleting row 1 and replacing row 2 with the sum of rows 1
and 2. Observe that column 1 of R⋆ has all zero entries, and the submatrix of R⋆ formed by columns 2f for 1 ≤ f ≤ n is 2R(2n−1 ). Therefore
P
1P
n
n−1 ) where E ⋆ = {1} ∪ {r/2 : r ∈ E\{1, 2} .
rc
r∈E R(2 )rc = 2
r∈E ⋆ R(2
By induction E ⋆ = {1, 2, . . . , 2n−1 }, and so E = D.
Part (ii) follows by a small extension of this argument. Let R⋆ be as
defined in (i). By Lemma 6.6, the entry of R⋆ in row r and column c is
odd if and only if r ∈ {p, 2p} and c = 2m where 0 ≤ m ≤ n. Any coprime
partition has a part containing both 2m and p for some such m. Therefore,
by parity, either both p and 2p are contained in E, or neither are. Deleting
row p and replacing row 2p with the sum of rows p and 2p of R⋆ , we obtain
2R(2n−1 p), augmented by two zero columns. The inductive argument for (i)
now shows that E = D.
Finally suppose that d = 2pn . Let R(2pn ) denote R(2pn ) with entries
regarded as elements of Z/pn Z. Let be the relation on D\{2pn } defined
as in (14), but working modulo pn . Let P E denote the set of equivalence
classes for . We need this preliminary result: if P E is coprime then 2, 2p,
. . . , 2pn ∈ E and P E has a single part. Again the proof is inductive. If
n = 1 then, by Lemma 6.6,


1
1
1
1
−1 −1 −1 −1


R(2p) = 

−1 −1 1
1
1
1 −1 −1
≃

≃

where the entries are in Z/pZ and D is ordered 1, p, 2, 2p. If 2p 6∈ E then,

since 1, 2 ∈ E, we have P E = {1, p}, {2, 2p} , which is not coprime.

Therefore 2p ∈ E and P E = {1, p, 2, 2p} , as required. Suppose that
⋆
n ≥ 2. Let R denote R(2pn ) with the entries taken in Z/pn−1 Z. Observe
⋆
that columns pn−1 and pn of R are equal, as are columns 2pn−1 and 2pn .
Moreover, rows pn and 2pn have all zero entries. By a very similar inductive
argument to (i), it follows that E contains 2, 2p,. . . , 2pn−1 . Let R⋆ be the
matrix obtained from R(pn ) by removing these rows, replacing row 2 with
their sum, and adding pe−1 to each entry in row pe , for 1 ≤ e ≤ n. For
example, if n = 3 then

R⋆ =


1 1
1
1
1 1
1
1

0 p
0 p
p
p
p
p



p 0
2
2
2
2
p
p
p 0
p
p



 2 2
3
2
2
3

p p
0
p
p p
0
p


 0 0 −p2
−p2
0 0 p2
p2 
0 0
p2 −p2 (p − 1) 0 0 −p2 p2 (p − 1)


where the row obtained by summation is highlighted. Since columns 1 and 2
of R⋆ are equal, and any part of a coprime partition of D\{2pn } contains
either 1 or 2, we see that P E has a single part. The column of R⋆ labelled

16

MARK WILDON

2pn−1 is greater, entry-by-entry, than every other column, except in rows pn
and 2pn . Since columns pn−1 and 2pn−1 of R⋆ are congruent except in the
summed row and row 2pn , and the sum of entries in these columns is less
than pn , we have 2pn ∈ E. This proves the preliminary result.
We now prove (ii). Each part of P E is a union of parts of PE , so P E
is coprime only if P E is coprime. By the preliminary result, 2, 2p, . . . ,
2pn ∈ E. Let R⋆⋆ be the matrix defined as R⋆ , but now adding all the rows
2, 2p, . . . , 2pn−1 , 2pn . The non-zero entries in the summed row for R⋆⋆ are
−pn in column pn and pn in column 2pn . Since pn is in a non-singleton part
of PE , we see from column pn that E contains 1, p, . . . , pn , as required. 
Despite its elementary statement, the author has been unable to prove
Conjecture 6.5 in any significantly greater generality. We offer this as an
open problem.
The Haskell [32] program RamanujanMatrix on the author’s website1
has been used to verify Conjecture 6.5 for all degrees d ≤ 600. We mention
that

 

R(pn ) =

1
1


1

 ..
.
1

0 0 ...
1 0 ...
1 1 ...
.. .. . .
.
. .
1 1 ...

0
0


0

.. 
.
1

−1

1
0


0

 ..
.
0

1 1
p p
0 p2
.. ..
. .
0 0

... 1
... p 


. . . p2  .

.
..
. .. 
. . . pn

It follows that each R(d) is invertible; the determinant of R(pn ) is pn(n+1)/2
and its inverse is R(pn )◦ /pn where R(pn )◦ is obtained from R(pn ) by rotation
by a half-turn. This leads to an alternative proof of Proposition 6.7(i) and
may be useful more widely.
7. Abelian B-groups
7.1. After CFSG. We now skip over many later developments, referring
the reader to Neumann’s essays in the collected works [7, 38] for some of the
missing history, and consider the situation after the Classification Theorem
of Finite Simple Groups. In an early application, it was used in [12] to determine all 2-transitive permutation groups. The resulting classification of all
primitive permutation groups containing a regular cyclic subgroup is given
in [14, Theorem 4.1] and [24, page 164], and independently refined in [23]
and [28]. We state the version of this result relevant to Theorem 1.1 below.
(Here Sd and Ad denote the symmetric and alternating groups of degree d,
respectively; the other notation is also standard.)
Theorem 7.1. Let G be a permutation group containing a regular cyclic
subgroup hgi of composite prime-power order pn . Then either G is imprimitive, or G is 2-transitive and one of the following holds:
1See www.ma.rhul.ac.uk/ uvah099/
~

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

17

(i) G = Apn or G = Spn and g is a pn -cycle;
(ii) PGLd (Fq ) ≤ G ≤ PΓLd (Fq ) where pn = (q d − 1)/(q − 1);
(iii) p = 3, n = 2, G = PΓL2 (F8 ) and g = sσ where s ∈ PGL2 (F8 )
is semisimple of order 3 and σ is the automorphism of PGL2 (F8 )
induced by the Frobenius twist.
Corollary 3 of [29] gives a rough classification of primitive permutation
groups containing a regular subgroup. This was sharpened by Li in [27,
Theorem 1.1] for regular abelian subgroups. Note that Case (2)(iv) of this
theorem, on groups with socle Sm × · · · × Sm or Am × · · · × Am , is missing
the assumption m ≥ 5. It is clear from Remark (b) following the theorem
and the structure of the proof in §5 that this assumption was intended; it
is required to exclude groups such as S2 ≀ Sr and A3 ≀ Sr with regular socle
whose product action is imprimitive. (Primitive groups such as S4 in its
natural action or S3 ≀ S2 in its product action are of affine type, and so
already considered in Case (1) of the theorem.)
It will be useful to say that a group K is m-factorizable if there exists
r ≥ 2 and groups K1 , . . . , Kr such that |K1 | = . . . = |Kr | = m and K ∼
=
K1 × · · · × Kr , and factorizable if it is m-factorizable for some m ≥ 3.
Proposition 7.2. If K is a regular abelian subgroup of a primitive but not
2-transitive permutation group G then either
∼ Fn is elementary abelian, the point stabiliser
(i) G = V ⋊ H where V =
p

H ≤ GL(V ) acts irreducibly on V but intransitively on V \{0} and
|K| = pn ; or
(ii) K is m-factorizable for some m ≥ 5.
Proof. If Case (1) of Li’s theorem applies then G ≤ AGLd (Fp ) where p is
prime and G acts on its socle V ∼
= Fdp . It is easy to show (see for example [13,
Theorem 4.8]) that G = V ⋊ H where H ≤ GL(V ) is irreducible. Since G is
not 2-transitive, H is not transitive. In the remaining case of Li’s theorem, G
is of the form (Te1 × · · · × Ter ).O.P where O ≤ Out(Te1 ) × · · · × Out(Ter ), P is
transitive of degree r and each Ter is an almost simple permutation group of
degree m ≥ 5. Moreover K = K1 × · · · × Kr where Ki < Tei and each Ki has
order m. Therefore, if r ≥ 2, then K is factorizable into m-subgroups with
m ≥ 5. If r = 1 then, as Li remarks following his theorem, G is 2-transitive,
so need not be considered any further.

Theorem 25.7 in [37] generalizes Example 6.3 to show that if m ≥ 3 and K
is m-factorizable with r factors then K is a regular subgroup of Sm ≀ Sr in
its primitive action on {1, . . . , k}r . This action is not 2-transitive, so K is
not a B-group. We therefore have the following corollary, first observed in
[27, Corollary 1.3].

18

MARK WILDON

Corollary 7.3. No factorizable group is a B-group. Moreover, an abelian
group not of prime-power order is a B-group if and only if it is not factorizable.
It is an open problem to determine the non-factorizable abelian B-groups
of prime-power order. We end with some partial results and reductions.
7.2. Elementary abelian B-groups. Exercise 3.5.6 in [13] asks for a proof
that Cpn is never a B-group. This is true when p > 2 by Corollary 7.3
(clearly Cp in its regular action is primitive but not 2-transitive), but false,
in general, when p = 2.2 For example, the primitive permutation groups of
degree 8 containing a regular subgroup isomorphic to C23 are A8 , S8 and the
affine groups F32 ⋊ C7 , F32 ⋊ (C7 ⋊ C3 ) and F32 ⋊ GL3 (F2 ). All of these groups
contain a 7-cycle, and so are 2-transitive. Therefore C23 is a B-group.
These examples motivate the following lemma, whose proof requires Burnside’s dichotomy on permutation groups of prime degree. The significance
of Mersenne primes will be seen shortly.
Lemma 7.4. Let V = Fn2 where 2n − 1 is prime. A subgroup H ≤ GL(V )
is transitive on V \{0} if and only if H ∼
= C2n −1 , H ∼
= C2n −1 ⋊ Cn or
H = GL(V ).
Proof. The ‘if’ direction is clear. By Theorem 6.1, if H is transitive on V \{0}
then either H ∼
= C2n −1 ⋊ Cr , for some r, or H is 2-transitive. Identifying
×
V \{0} with F2n , we see that there exists h ∈ GL(V ) of order 2n − 1. (Such
elements are called Singer cycles.) Let α be a primitive (2n − 1)-th root of
unity. Note that h is conjugate to hs in GL(V ) if and only if the map β 7→ β s
n−1
of h. Thus NGL(V ) (hhi) ∼
permutes the eigenvalues α, α2 , . . . , α2
= Cn is
2
generated by an element of prime order n conjugating h to h , and either
r = 1 or r = n. If H is 2-transitive then V ⋊ H is 3-transitive. Such groups
were classified by Cameron and Kantor in [9]. By their Theorem 1 in the
case of vector spaces over F2 , the only such groups are V ⋊GL(V ) and, when
n = 4, V ⋊ A7 . Since 24 − 1 is composite, only the former case arises.

It is worth noting that [9] predates the classification theorem; the methods
used are mainly from discrete geometry rather than group theory. More
generally, Hering [16, 17] has classified the linear groups H transitive on nonzero vectors, under various assumptions on the composition factors of H.
Proposition 7.5. Let V = Fn2 . The elementary abelian group C2n is a
B-group if and only if 2n − 1 is a Mersenne prime and the only simple
irreducible subgroups of GL(V ) are C2n −1 and GL(V ).
2This

mistake
is
corrected
in
people.math.carleton.ca/~ jdixon/Errata.pdf.

the

errata

available

at

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

19

Proof. Suppose that 2n −1 is composite. Let h ∈ GL(V ) be a Singer cycle. If
n 6= 6 then, by Zsigmondy’s Theorem [39], there exist a prime r such that r
divides 2n − 1 and r does not divide 2m − 1 for any m < n. Thus hn/r does
not permute the vectors of a non-zero proper subspace of V , and so hhn/r i
acts irreducibly on V and intransitively on V \{0}. Therefore V ⋊ hhn/r i is
primitive but not 2-transitive, and so C2n is not a B-group. In the exceptional
case of Zsigmondy’s Theorem when n = 6, we simply take h3 , of order 21.
Suppose that 2n − 1 is prime and that there is a simple irreducible group
T ≤ GL(V ) other than C2n −1 and GL(V ). By Lemma 7.4, T is intransitive
on V \{0}, and so V ⋊ T is not 2-transitive. Hence C2n is not a B-group.
Conversely, assume that no such simple group exists, and, for a contradiction, that C2n is not a B-group. By Proposition 7.2, there exists a proper
irreducible subgroup H of GL(V ) such that H is intransitive on V \{0}.
Let M be a maximal subgroup of GL(V ) containing H. The maximal subgroups of classical groups were classified by Aschbacher in [1]. Of the 11
Aschbacher classes, the first consists of reducible groups, and the remaining 10 of groups preserving a structure on V that can exist only when V has
composite dimension. Therefore M is an almost simple group. Since M is
a proper subgroup of GL(V ), Lemma 7.4 implies that M is intransitive on
V \{0}. Let T be the simple normal subgroup of M . By Clifford’s Theorem
([11, Theorem I]), the restriction of V to T decomposes as a direct sum of
irreducible representations of T of the same dimension. Since n is prime, T
acts irreducibly on V . Its orbits are contained in the orbits of M , so it acts
intransitively on V \{0}, contrary to our assumption.

By Proposition 7.5, a solution to the following problem will imply that C2n
is a B-group if and only if 2n − 1 is a Mersenne prime.
Problem 7.6. Show that if 2n − 1 is a Mersenne prime and n ≥ 3 then
no non-abelian finite simple group other than GLn (F2 ) has an irreducible
representation of dimension n over F2 .
The two remarks below give some partial progress on Problem 7.6.
(1) The Atlas [22] data available in gap [15] shows that, with the possible exceptions of J4 , Ly, T h, F i24 , B and M , no sporadic simple group has an irreducible representation over F2 of dimension n
where 2n − 1 is a Mersenne prime. Indeed, it appears to be rare for a
sporadic group or a finite group of Lie type to have a non-trivial irreducible representation over F2 of odd dimension. The author knows
of no examples of such representations of alternating groups. Since
a self-dual representation has an invariant alternating form, whereas
an odd-dimensional orthogonal group over F2 has a 1-dimensional
invariant subspace, such a representation is necessarily not self-dual.

20

MARK WILDON

(2) Inspection of tables of small dimensional representations of quasisimple groups [18, 19] and (for the groups deliberately excluded therein),
Chevalley groups in defining characteristic [30] show that no finite
simple group except for GLn (F2 ) has an irreducible representation
over F2 of dimension n ≤ 250 such that 2n − 1 is a Mersenne prime.
Thus if n ≤ 250 then C2n is a B-group if and only if
n ∈ {1, 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127}.
7.3. Non-elementary abelian B-groups. An interesting feature of the
affine groups in Proposition 7.2(i) is that they may contain regular abelian
subgroups other than Cpn . In Remark 1.1 in [27], Li gives the example
V ⋊ Sn where V is the subrepresentation he2 − e1 , . . . , en − e1 i of the natural
permutation representation he1 , . . . , en i of Sn over Fp . To avoid a potential
ambiguity, let tv ∈ V ⋊ H denote translation by v ∈ V . If 2s < n then the
subgroup of V ⋊ H generated by
(2, 3)te1 +e2 , . . . , (2s, 2s + 1)te1 +e2s , te2s+2 , . . . , ten
is regular and isomorphic to C4s ×C2n−2s−1 . Li claims that V ⋊H is primitive.
However H acts irreducibly only when n is odd (and so dim V is even, as
expected by Remark (1) above). Thus if r ∈ N0 and s ∈ N then C4s × C22r
is not a B-group, but Li’s example sheds no light on when C4s × C22r+1 ,
which may be non-factorizable, is a B-group. This is a special case of the
following problem.
Problem 7.7. Classify non-elementary abelian B-groups of prime-power
order.
By Proposition 7.2, this problem reduces to classifying regular abelian
subgroups of affine groups V ⋊ GL(V ). The main result of [10] is a beautiful bijective correspondence between such subgroups and nilpotent algebras
with underlying vector space V . To explain part of this correspondence, observe that if K is an regular abelian subgroup of V ⋊ H where H ≤ GL(V )
then, for each v ∈ V , there exists a unique hv ∈ H such that hv tv ∈ K.
From hu hv tuhv +v = hu tu hv tv = hv tv hu tu = hv hu tvhu +u for u, v ∈ V , we see
that {hv : v ∈ V } is an abelian subgroup of H and uhv + v = vhu + u for
all u, v ∈ V . Replacing v with v + w, we obtain
uhv+w + (v + w) = (v + w)hu + u = vhu + whu + u = uhv + v + uhw + w − u
and so, cancelling v + w and subtracting u, we have
(15)

hv+w − 1 = (hv − 1) + (hw − 1)

for all v, w ∈ K. This additivity property is highly restrictive.

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

21

Example 7.8. Let K = {hv tv : v ∈ V } be a regular abelian subgroup of
V ⋊ Sn , where V is as in Li’s example. The matrix X representing hv in the
basis e2 − e1 , . . . , en − e1 of V is a permutation matrix if and only if 1hv = 1.
If 1hv = a and bhv = 1 then, since (ei − e1 )hv = (eihv − 1) − (ea − 1), each
entry of X in column ea − e1 is −1, row ei − e1 has a 1 in column eihv − e1 for
each i 6= b, and X has no other non-zero entries. By (15), h2v = 2hv − 1, so
h2v is represented by 2X − I, where I is the identity matrix. But 2X − I is
not of either of these forms unless p = 2 or X = I. Therefore V is the unique
regular abelian subgroup of V ⋊ Sn if p > 2. Suppose that p = 2. If hv has
order 4 or more, the matrix representing hv + h−1
v + 1 has multiple non-zero
entries in the columns for both ea − e1 and eb − e1 , again contradicting (15).
Therefore each hv has order at most 2. It follows that K has exponent 2
or 4. Thus the examples given by Li are exhaustive.
When p divides n the representation V has an irreducible quotient U =
V /he1 + · · · + en i. Similar arguments show that U ⋊ Sn has a non-elementary
abelian regular subgroup if and only if p = 2. Any such subgroup has
exponent 4, with the exception that when n = 6, U ⋊ S6 has an regular
abelian subgroup isomorphic to C8 × C2 . This does not contradict the result
first claimed by Manning (see §6.3) since in this case S6 acts transitively
on U \{0}; the related 2-transitive action of A7 on F42 , coming from the
isomorphism A8 ∼
= GL4 (F2 ), was seen in the proof of Lemma 7.4.
We end with some consequences of the following observation: if J is the
r
m × m unipotent Jordan block matrix over Fp then J p = I if and only if
r
pr ≥ m and I + J + · · · + J p −1 = 0 if and only if pr > m. (The latter can
r
r
be proved most simply using the identity I + J + · · · + J p −1 = (J − I)p −1 .)
Proposition 7.9. Let V = Fnp and let K be a regular abelian subgroup of
V ⋊ GL(V ).
(i) If n < p then K ∼
= Cpn .
∼
(ii) If K = Cpn then either n = 1 or p = 2 and n = 2.
r

Proof. For hv tv ∈ K we have (hv tv )p = hpv tw where w = v + vhv + · · · +
r
vhpv −1 . Hence, using the observation just made, if n < p then (hv − 1)p = 0
and so hpv = 1 and (hv tv )p = 1, giving (i). Now suppose that hv tv genern−1
n−1
6= 1, we have v + vhv + · · · + vhpv −1 6= 0. Hence
ates K. Since (hv tv )p
there is a m × m unipotent Jordan block in hv with m ≥ pn−1 . Therefore
n ≥ pn−1 which implies (ii).

r

The subgroups K in Proposition 7.9(i) may be classified up to conjugacy
in the affine group using the theory in [10]. Using Proposition 7.9(i) to
rule out degrees, it follows from an exhaustive search through the library
of primitive permutation groups in magma [2] that the abelian B-groups of
composite prime-power degree d where d ≤ 255 are precisely those listed

22

MARK WILDON

d

pn

f (d) abelian B-groups of order d

4
8
9
16
25
27
32
49
64
81
121
125
128

22
23
32
24
52
33
25
72
26
34
112
53
27

0
0
2
9
17
9
0
29
55
125
43
38
0

169 132 64
243 35 30

C22 , C4
C23 , C4 × C2 , C8
C9
C8 × C2 , C16
C25
C9 × C3 , C27
C25 , C4 × C23 , C42 × C2 , C8 × C22 , C8 × C4 , C16 × C2 , C32
C49
C16 × C22 , C16 × C4 , C32 × C2 , C64
C27 × C3 , C81
C121
C25 × C5 , C125
C27 , C4 × C25 , C42 × C23 , C4 × C25 , C8 × C24 , C8 × C4 × C22 , C8 × C42 ,
C82 × C2 , C16 × C23 , C16 × C4 × C2 , C16 × C8 , C32 × C22 , C32 × C4 ,
C64 × C2 , C128
C169
C9 × C33 , C9 × C9 × C3 , C27 × C32 , C27 × C9 , C81 × C3 , C243

Table 1. All abelian B-groups of composite prime-power
degree d where d ≤ 255; f (d) is the number of primitive
permutation groups of degree d that are not 2-transitive.

in Table 1 above. Finally we remark that Proposition 7.2 and Proposition 7.9(ii) together imply that Cpn is a B-group for all primes p and all
n ∈ N with n ≥ 2, giving one final proof of Theorem 1.1.
Acknowledgements
I thank Nick Gill for his answer to a MathOverflow question outlining
an alternative proof of Proposition 7.2 and Derek Holt for several helpful
comments on this question (see mathoverflow.net/questions/258434/).
I thank John Britnell and Peter M. Neumann for helpful comments.
References
[1] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math.
76 (1984), 469–514.
[2] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I.
The user language, J. Symbolic Comput. 24 (1997), 235–265, Computational algebra
and number theory (London, 1993).
[3] W. Burnside, On some properties of groups of odd order, Proc. London Math. Soc.
33 (1901), 162–185.
On simply transitive groups of prime degree, Q. J. Math. 37 (1906), 215–221.
[4]

PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP

[5]
[6]
[7]

[8]

[9]
[10]
[11]
[12]

[13]
[14]

[15]
[16]
[17]
[18]
[19]
[20]

[21]
[22]

[23]
[24]

[25]

23

On certain simply-transitive permutation-groups, Proc. Cambridge Phil. Soc.
20 (1921), 482–484.
Theory of groups of finite order, Dover Publications Inc., New York, 1955,
reprint of 2nd edition, Cambridge University Press, 1911.
The collected papers of William Burnside. Vol. I, Oxford University Press,
Oxford, 2004, Commentary on Burnside’s life and work; papers 1883–1899, edited
by Peter M. Neumann, A. J. S. Mann and Julia C. Tompson and with a preface by
Neumann and Mann.
The collected papers of William Burnside. Vol. II, Oxford University Press,
Oxford, 2004, papers 1900–1926, edited by Peter M. Neumann, A. J. S. Mann and
Julia C. Tompson and with a preface by Neumann and Mann.
P. J. Cameron and W. M. Kantor, 2-transitive and antiflag transitive collineation
groups of finite projective spaces, J. Algebra 60 (1979), 384–422.
A. Caranti, Francesca Dalla Volta, and Massimiliano Sala, Abelian regular subgroups
of the affine group and radical rings, Publ. Math. Debrecen 69 (2006), 297–308.
A. H. Clifford, Representations induced in an invariant subgroup, Ann. of Math. (2)
38 (1937), 533–550.
Charles W. Curtis, William M. Kantor, and Gary M. Seitz, The 2-transitive permutation representations of the finite Chevalley groups, Trans. Amer. Math. Soc. 218
(1976), 1–59.
John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996.
Walter Feit, Some consequences of the classification of finite simple groups, The Santa
Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc.
Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 175–181.
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.4, 2014.
C. Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geometriae Dedicata 2 (1974), 425–460.
Transitive linear groups and linear groups which contain irreducible subgroups
of prime order. II, J. Algebra 93 (1985), 151–164.
Gerhard Hiss and Gunter Malle, Low-dimensional representations of quasi-simple
groups, LMS J. Comput. Math. 4 (2001), 22–63.
Corrigenda: “Low-dimensional representations of quasi-simple groups”, LMS
J. Comput. Math. 5 (2002), 95–126.
Gordon James and Adalbert Kerber, The representation theory of the symmetric
group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley
Publishing Co., Reading, Mass., 1981.
G. D. James, The representation theory of the symmetric groups, Lecture Notes in
Mathematics, vol. 682, Springer, Berlin, 1978.
Christoph Jansen, Klaus Lux, Richard Parker, and Robert Wilson, An atlas of Brauer
characters, London Mathematical Society Monographs. New Series, vol. 11, The
Clarendon Press, Oxford University Press, New York, 1995, Appendix 2 by T. Breuer
and S. Norton, Oxford Science Publications.
Gareth A. Jones, Cyclic regular subgroups of primitive permutation groups, J. Group
Theory 5 (2002), 403–407.
William M. Kantor, Some consequences of the classification of finite simple groups,
Finite groups—coming of age (Montreal, Que., 1982), Contemp. Math., vol. 45, Amer.
Math. Soc., Providence, RI, 1985, pp. 159–173.
Wolfgang Knapp, On Burnside’s method, J. Algebra 175 (1995), 644–660.

24

MARK WILDON

[26] Rudolf Kochendörffer, Untersuchungen über eine Vermutung von W. Burnside,
Schriften des mathematischen Seminars und des Instituts für angewandtemathematik
der Universität Berlin. 3 (1937), 155–180.
[27] Cai Heng Li, The finite primitive permutation groups containing an abelian regular
subgroup, Proc. London Math. Soc. (3) 87 (2003), 725–747.
[28]
Permutation groups with a cyclic regular subgroup and arc transitive circulants, J. Algebraic Combin. 21 (2005), 131–136.
[29] Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl, Transitive subgroups of primitive
permutation groups, J. Algebra 234 (2000), 291–361.
[30] Frank Lübeck, Small degree representations of finite Chevalley groups in defining
characteristic, LMS J. Comput. Math. 4 (2001), 135–169.
[31] Dorothy Manning, On simply transitive groups with transitive abelian subgroups of
the same degree, Trans. Amer. Math. Soc. 40 (1936), 324–342.
[32] Simon Peyton Jones et al., The Haskell 98 language and libraries: The
revised report, Journal of Functional Programming 13 (2003), 0–255,
http://www.haskell.org/definition/.
[33] S. Ramanujan, On certain trigonometrical sums and their applications in the theory
of numbers, Trans. Cambridge Philos. Soc. 22 (1918), 259–276.
[34] I. Schur, Neuer Beweis eines Satzes von W. Burnside, Jahresbericht der Deutschen
Mathematik-Vereinigung 17 (1908).
[35] Lawrence C. Washington, Introduction to cyclotomic fields, second ed., Graduate
Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997.
[36] Helmut Wielandt, Zur Theorie der einfach transitiven Permutationsgruppen, Math.
Z. 40 (1936), 582–587.
[37]
Finite permutation groups, Translated from the German by R. Bercov, Academic Press, New York-London, 1964.
[38]
Mathematische Werke/Mathematical works. Vol. 1, Walter de Gruyter & Co.,
Berlin, 1994, Group theory, With essays on some of Wielandt’s works by G. Betsch,
B. Hartley, I. M. Isaacs, O. H. Kegel and P. M. Neumann, edited with a preface by
Bertram Huppert and Hans Schneider.
[39] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), 265–
284.

