arXiv:1602.00231v1 [math.RA] 31 Jan 2016

The Multiplicative Automorphisms of a Finite
Nearfield, with an Application
Tim Boykett and Karin-Therese Howell
February 2, 2016
Abstract
In this paper we look at the automorphisms of the multiplicative group
of finite nearfields. We find partial results for the actual automorphism
groups. We find counting techniques for the size of all finite nearfields.
We then show that these results can be used in order to count the number
of near vector spaces of a given dimension over a given nearfield, up to
isomorphism.

1

Introduction

In this paper we look at the automorphisms of the multiplicative group of finite
nearfields. While these are easily found explicitly for fields and the seven exceptional nearfields, we only obtain explicit structural results for some classes
of Dickson nearfields.
In the next section we look at finite dimensional near vector spaces. We show
that the results about the multiplicative group automorphisms can be used in
order to count the number of near vector spaces of a given dimension over a
given nearfield, up to isomorphism.
In the last section we obtain results allowing us to count the size of the
automorphism group of a Dickson nearfield, allowing us to calculate the number
of near vector spaces explicitly.

2

Background

A nearfield is a (2, 2)-algebra (N, +, ∗) such that (N, +) is a group with identity
0, (N \ {0}, ∗) is a group and one distributive law applies, in our case the
right distributive law, (a + b) ∗ c = a ∗ c + b ∗ c for all a, b, c ∈ N . Clearly
fields are examples of nearfields. We will often omit the ∗ symbol and notate
multiplication by juxtaposition. The study of nearfields arose from an interest
in the axiomatics of fields. Dickson showed that proper nearfields that are
not fields exist using a construction that now bears his name, which we will
encounter later.
Nearfields have found important applications in combinatorics and are used
in the study of finite geometries, coordinatising certain types of finite planes.
Sharply 2-transitive group actions can be shown to be affine maps x 7→ ax + b

1

on nearfields in the finite case. It has recently been shown that there are nonnearfield based examples in the infinite case [11].
By Zassenhaus [15], here are three classes that a finite nearfield N can belong
to:
1. N is a finite field,
2. N is a Dickson nearfield,
3. N is one of 7 exceptional nearfields.
In all cases the additive group of a nearfield is elementary abelian. Other
facts about nearfields will be introduced below.
We will use ϕ(n) for the Euler Totient function. We will use Zn for the
cyclic group of order n, as well as the integers modulo n. The context should
help remove any ambiguity.

3

Automorphism groups of finite nearfields

We are interested in exploring the multiplicative structure of finite nearfields
through the automorphisms of the group. For nearfields that are fields, this is
the automorphism group of a cyclic group. The rest of this section will be spent
determining the multiplicative automorphism group for those finite nearfields
that are not fields.

3.1

Dickson Nearfields

A Dickson nearfield is a “twisting” of a field. The twisting is defined by a
Dickson Pair.
Definition 1 A pair of numbers (q, n) ∈ N2 is called a Dickson pair if q is
some power pl of a prime p and each prime divisor of n divides q − 1 and
4|n ⇒ 4|(q − 1).
All Dickson nearfields arise by taking Dickson pairs as described in Theorem
8.31, p. 244 in [10].
For any Dickson pair (q, n) we will denote the associated Dickson nearfield
by DF (q, n), the multiplicative group by Gq,n . The group is metacyclic and can
be presented as
Gq,n = a, b | am = 1, bn = at , ba = aq b
where t = m/(q − 1) and m = (q n − 1)/n. [14, p 168 (IV.1.1.d)].
For ease of calculation, we will generally write elements in the form ai bj . We
note that
j−1
(ak b)j = ak(q +...+q+1) bj
by the third equation above.
The following result gives us some structure of the group Gq,n , using the
numbering of the book, but different notation.
Theorem 2 ([14] Theorem IV.1.5) Let (q, n) be a Dickson pair, m = (q n −
1)/n and t = m/(q − 1).
2

(a) Z(Gq,n ) = hat i
(b) The Sylow subgroups of Gq,n are cyclic or quaternion.
(c) gcd(n, t) = gcd(q − 1, t) ≤ 2 and the following are equivalent:
(I) The 2-Sylow subgroups are cyclic
(II) gcd(q − 1, t) = 1
(III) n is odd or q 6≡ 3 (mod 4)
(d) hai is a maximal abelian subgroup of Gq,n
(e) If (q, n) 6= (3, 2) then hai includes all abelian normal subgroups of Gq,n . In
particular, hai is characteristic in Gq,n .
The automorphisms of a Dickson nearfield are determined in the following
main result. The multiplicative group G3,2 is the quaternion group, which has
24 automorphisms, isomorphic to S4 .
n

Theorem 3 Let (q, n) be a Dickson pair, (q, n) 6= (3, 2), with m = q n−1 and
m
. Then φ ∈ Aut(Gq,n ) iff there exist some i, k ∈ Zm such that φ(a) = ai
t = q−1
and φ(b) = ak b, with gcd(i, m) = 1 and solving
k

qn − 1
+ t ≡ it
q−1

(mod m)

(1)

Proof: (⇐) Every map of the generators into Gq,n that respects the equations
in the presentation can be extended to an endomorphism.We use φik to denote
the morphism with φik (a) = ai and φik (b) = ak b. Firstly we show that the
images satisfy the three equations of the presentation.
φik (a)m = aim = 1
n

k

n

(2)

φik (b) = (a b) = a

k(1+q+...+qn−1 ) n

= ak(1+q+...+q
it

b

n−1

)+t

(4)

t

= a = φik (a)
k

i

k iq

(3)
(5)

iq k

q

φik (b)φik (a) = a ba = a a b = a a b = φik (a) φik (b)

(6)

Now we show that the homomorphism satisfying these requirements is a
bijection. Suppose aα bβ ∈ ker φik . Then
φik (aα bβ ) = 1 ⇔ aαi (ak b)β = 1
⇔ aαi ak(1+q+...+q

(7)
β−1

) β

b = 1.

(8)

Thus β = 0, so φik (aα ) = 1 ⇔ aαi = 1. Because i is coprime to m, this implies
that α = 0 so the kernel of φik is trivial and we have an automorphism.
(⇒) We show the form of the automorphisms by investigating the images of
the generators under the automorphism. If mapping of the generators a, b into
Gq,n is an automorphism then the images satisfy the same equations.
By Theorem 2 we know that the subgroup generated by a is characteristic,
so a must map to some power of itself. In order for φ to be an automorphism,
this power must be coprime to the order of a, which is m, since am = 1.
3

In general, φ(b) = ak bj for some j, k, since ba = aq b. The images of a and b
must satisfy the same equations as a and b, so ba = aq b requires
(ak bj )ai = aiq ak bj
k iqj j

⇔a a

(9)

k iq j

b =a a b

(10)

j

⇔ aiq = aiq
j

⇔ iq ≡ iq
j

⇔q ≡q

(11)
(mod m)

(12)

(mod m) because i is coprime to m

(13)

Let α = j − 1 and we have the condition q α ≡ 1 (mod m) or equivalently,
m|q α − 1. Taking logarithms to the base q, we obtain
logq m ≤ logq (q α − 1) ≤ α and
qn − 1
logq m = logq
= logq (q n − 1) − logq n > n − 1 − logq n
n

(14)
(15)

so α > n − 1 − logq n.
The set A = {α ∈ Zn | q α ≡ 1 (mod m)} ≤ (Zn , +) as a group, and 0 6= α ∈
A ⇒ n − α ∈ A. Note also that A is always a proper subgroup, as 1 6∈ A.
Suppose A is not trivial, so n ≥ 4. Then there is some α ∈ A, α > n − 1 −
logq n. So n − α < n − (n − 1 − logq n) = 1 + logq n. But also n − α ∈ A, so
n − α > n − 1 − logq n, so 1 + logq n > n − 1 − logq n, that is, 2 logq n > n − 2 so
2
n2 > q n−2 . This can be rewritten as q < n n−2 . For n = 4, the upper bound is
2
n n−2 = 4 and the bound is monotone descending. There are no Dickson pairs
(q, n) with q < 4 and n ≥ 4. Thus A is always trivial, so we know that j = 1.
Thus we know that φ(b) = ak b for some k ∈ Zm .
The second equation in the presentation then requires that
(ak b)n = (ai )t
⇔a

k(qn−1 +...+q+1) n

⇔ ak(q
⇔ k(q

n−1

(17)

a = ait

(18)

b =a

n−1

(16)

it

+...+q+1) t

+ . . . + q + 1) + t ≡ it

(mod m)

(19)

which is the condition we wanted.



The structure of the automorphism group is a semidirect product.
Lemma 4 Let U(Zm ) denote the group of units of Zm under multiplication.
Then Aut(Gq,n ) ≤ U(Zm ) ⋊ Zm , with the units acting by multiplication.
Proof: Let (i, k), (j, l) ∈ U(Zm ) × Zm and define (i, k) ∗ (j, l) = (ij, k + il).
This gives a multiplication making (U(Zm ) × Zm , ∗) a group that is a semidirect
product of the units of Zm with Zm .
Let φik , φjl ∈ Aut(Gq,n , ∗). Then
φik ◦ φjl (a) = φik (aj ) = aij

(20)

l

l

il k

φik ◦ φjl (b) = φik (a b) = φik (a )φik (b) = a a b = a

4

k+il

b

(21)

so φik ◦ φjl = φ(ij)(k+il) . Clearly this is the multiplication defined in the semidirect product above, so the automorphisms form a subgroup of this semidirect
product.

We note that the inner automorphisms are easily found.
Lemma 5 Let (q, n) be a Dickson pair. Then the inner automorphisms of Gq,n
are of order tn.
Proof: The center of the nmultiplicative
group of a Dickson nearfield is of order
n
−1
n
m
= q n−1 q−1
= q−1
n = tn.

q − 1, so |Inn(Gq,n )| = qq−1
We can determine the structure of the automorphism group directly in some
cases.
Definition 6 ([6]) Let m, n, k ∈ N such that k n ≡ 1 (mod m). Then D(m, n; k)
is the metacyclic group of order mn defined by the presentation
hx, y|xm = y n = 1, yxy −1 = xk i.

(22)

For m, n coprime, there exists an explicit description of the automorphism
group of D(m, n; k).
Proposition 7 ([5] Proposition 1.3) Let A and G be finite groups of relatively prime orders, A cyclic with a G-action α. Then
Aut(A ×α G) ≡ A/AG ×∗ (Aut(A) × Autα (G))
with Autα (G) = {φ ∈ Aut(G)|α = αφ}, AG = {a ∈ A|ag = a for all g ∈ G}
and action (φ1 , φ2 ) ∗ x = φ1 (x) for all (φ1 , φ2 ) ∈ Aut(A) × Autα (G) and x ∈
A/AG .
Theorem 8 Let (q, n) be a Dickson pair with n odd or q 6≡ 3 (mod 4). Then
Gq,n ∼
= D(r, s; q r̄ ) with r = tr̄, s = ns̄ and q = r̄s̄ + 1
Proof: The order of Gq,n is q n − 1 = mn = t(q − 1)n. By Theorem 2 gcd(n, t) =
gcd(q − 1, t) = 1. Let r̄s̄ = q − 1 such that all prime factors of n are in s̄,
gcd(r̄, n) = 1. Then define r = tr̄ and s = ns̄ so rs = q n − 1.
Let a, b be the generators of Gq,n . Define ā = as̄ , b̄ = br̄ . Then

b̄āb̄

−1

ār = as̄r = as̄tr̄ = at(q−1) = am = 1

(23)

b̄s = br̄s = br̄s̄n = atr̄s̄ = at(q−1) = 1

(24)

r̄ s̄ −1

= b a b̄

=a

r̄qr̄ −1

b̄

qr̄

= ā b̄b̄

−1

qr̄

= ā .

(25)

Thus A = hā, b̄i ≤ Gq,n is a homomorphic image of D(r, s; q r̄ ).
We now show that A = Gq,n , thus of the same order as D, thus isomorphic.
By calculation, b̄n = atr̄ = ar and gcd(r, s̄) = 1 so a ∈ har , as̄ i. Since gcd(r̄, n) =
1 there exists some j such that r̄j ≡ 1 (mod n) so b̄j = bβn+1 = aβt b. Since
a ∈ A then aβt ∈ A and so a, b ∈ A, A = Gq,n and we are done.

Thus we can use Proposition 7 to determine the structure of the automorphism group in this case. We have not been able to obtain explicit descriptions
of the automorphism group, or any description of the automorphism group in
the case that gcd(n, t) = 2. However, in Corollary 18 below we will obtain an
explicit size of the automorphism group for all finite Dickson nearfields.
5

3.2

Exceptional Nearfields

The nearfield multiplicative groups are given in [14, Kapital IV], allowing us to
simply calculate the values needed. The following facts are collected from [14,
(IV.7.1),(IV.8.1)] with calculations in [1, 4].
Name
(I)52
(II)112
(III)72
(IV )232
(V )112
(V I)292
(V II)592

Mult Gp
SL(2, 3)
SL(2, 3) × Z5
2O
2O × Z11
SL(2, 5)
SL(2, 5) × Z7
SL(2, 5) × Z29

Group Aut
S4
S4 × Z4
S4 × Z2
S4 × Z2 × Z10
S5
S5 × Z6
S5 × Z28

We use 2O to represent the binary octahedral group, SL(n, q) the special
linear group of dimension n over the finite field of order q.

4

An application to counting near vector spaces

In [2], the concept of a vector space, i.e., linear space, is generalized by André to
a structure comprising a bit more non-linearity, the so-called near vector space.
In [13] van der Walt showed how to construct an arbitrary finite-dimensional
nearvector space, using a finite number of nearfields, all having isomorphic multiplicative semigroups. In [9] this construction is used to characterize and count
all finite-dimensional nearvector spaces over arbitrary finite fields.
Our aim is to use our results to count the number of non-isomorphic nearvector spaces over a given nearfield.

4.1

Near vector spaces

We begin with a brief overview and some definitions. See [2] for further details.
Definition 9 ([2]) A pair (V, A) is called a near vector space if:
1. (V, +) is a group and A is a set of endomorphisms of V ;
2. A contains the endomorphisms 0, id and −id;
3. A∗ = A \ {0} is a subgroup of the group Aut(V );
4. A acts fixed point free (fpf ) on V , i.e., for x ∈ V, α, β ∈ A, xα = xβ
implies that x = 0 or α = β;
5. the quasi-kernel Q(V ) of V , generates V as a group. Here, Q(V ) = {x ∈
V | ∀α, β ∈ A, ∃γ ∈ A such that xα + xβ = xγ}.
We sometimes refer to V as a near vector space over A. The elements of V
are called vectors and the members of A scalars. The action of A on V is called
scalar multiplication. Note that −id ∈ A implies that (V, +) is an abelian group.
Also, the dimension of the near vector space, dim(V ), is uniquely determined
by the cardinality of an independent generating set for Q(V ).
6

In [13](Theorem 3.4, p.301) van der Walt derives a characterization of finite
dimensional nearvector spaces:
Theorem 10 Let V be a group and let A := D ∪ {0}, where D is a fix point free
group of automorphisms of V . Then (V, A) is a finite dimensional near vector
space if and only if there exists a finite number of nearfields, F1 , F2 , . . . , Fn ,
semigroup isomorphisms ψi : A → Fi and a group isomorphism Φ : V →
F1 ⊕ F2 ⊕ · · · ⊕ Fn such that if
Φ(v) = (x1 , x2 , · · · , xn ), (xi ∈ Fi )
then
Φ(vα) = (x1 ψ1 (α), x2 ψ2 (α), · · · , xn ψn (α)),
for all v ∈ V and α ∈ A.
According to this theorem we can specify a finite dimensional near vector
space by taking n nearfields F1 , F2 , . . . , Fn for which there are semigroup isomorphisms ϑij : (Fj , ·) → (Fi , ·) with ϑij ϑjk = ϑik for 1 ≤ i, j, k ≤ n. We can
then take V := F1 ⊕ F2 ⊕ · · · ⊕ Fn as the additive group of the near vector space
and any one of the semigroups (Fio , ·) as the semigroup of endomorphisms by
defining
(x1 , x2 , . . . , xn )α := (x1 ϑ1io (α), x2 ϑ2io (α), · · · , xn ϑnio (α)),
for all xj ∈ Fj and all α ∈ Fio .
Definition 11 Two near vector spaces V and W over the same nearfield N
are isomorphic if there is a group isomorphism θ : (V, +) → (W, +) such that
θ(vn) = θ(v)n for all v ∈ V and n ∈ N .
Finite near vector spaces have a finite dimension and a finite base nearfield.
Thus we can represent a finite near vector space over N as a finite sequence of
multiplicative automorphisms of N , at least one of which is the identity. An
isomorphism between two such representations (α1 , . . . , αn ) and (β1 , . . . , βn ) is
a permutation γ of {1, . . . , n} and a collection of nearfield automorphisms δi
such that
αi = δi βγi
Lemma 12 An isomorphism class of near vector spaces of dimension n over a
nearfield N is a multiset of size n of cosets of Aut(N, +, ·) in Aut(N, ·), at least
one of which is Aut(N, +, ·).
Proof: A multiset is a sequence modulo permutations. So the γ above is identity.
Then the condition above is
αi βi−1 ∈ Aut(N, +, ·)

Let us order the cosets as K1 , K2 , . . . , Kk with K1 = Aut(N, +, ·). Then an
isomorphism class representative can be written as a nondecreasing sequence of
7

length n, starting with 1, of elements in 1, . . . , k. This number is k multichoose
n − 1, or


n+k−2
n
by the counting result for multisets [12, Theorem 5.3.2].
The rest of this paper will be concerned with calculating the factor k in the
above result.

5

Counting

This section will be concerned with calculating the size of the automorphism
group of a nearfield. We will look at three cases corresponding to Zassenhaus’
result. We will call the factor F (N ), the index of the nearfield automorphism
group within the multiplicative automorphism group of a nearfield N .

5.1

Fields

Suppose N = GF (q) is a finite field of order q = pn . Then the field automorphisms are generated by the Frobenius automorphism x 7→ xp and so the size
of the automorphism group is n. The multiplicative group of a finite field is
cyclic of order q − 1, so there are ϕ(q − 1) generators. Thus there are ϕ(q − 1)
multiplicative automorphisms and the index of the nearfield automorphisms in
the multiplicative automorphisms is
F (GF (q)) =

ϕ(q − 1)
n

This agrees with the result in [8, Theorem 3.2] for prime fields and [9, Theorems
3.8 and 3.9] in general.

5.2

Exceptional Nearfields

The nearfield multiplicative groups and their automorphism groups are given in
[14, Kapital IV], allowing us to simply calculate the values needed. The following
facts are collected from [14, (IV.7.1),(IV.8.1)] with calculations in [1, 4].
Name
(I)52
(II)112
(III)72
(IV )232
(V )112
(V I)292
(V II)592

Nearfield Aut
4
2
3
1
5
2
1

Group Aut Factor
24
6
96
48
48
16
480
480
120
24
720
360
3360
3360

So we are done here, and turn our attention to the case of Dickson nearfields.

8

5.3

Counting for Dickson Nearfields

Let (q, n) be a Dickson pair, with q = pl . The nearfield automorphism group has
order 6 for (3,2), otherwise it has order ln/k where k is the order of p (mod n)
[14, Theorem (2.3), p. 175].
Our problem now is to count the number of solutions to the equations in
Theorem 3 for a given Dickson pair.
Definition 13 Let (q, n) be a Dickson pair. Define
S(q, n) = {(i, k) ∈ Zm × Zm | i, k satisfies the equations in Theorem 3}
and
T (q, n) = {(i, k) ∈ Zm × Zq−1 | gcd(i, m) = 1, kn ≡ i − 1 (mod q − 1)}
with m =

qn −1
n .

Lemma 14 Let (q, n) be a Dickson pair. Then |S(q, n)| = t|T (q, n)| with t =
m
q−1 .
Proof: The equation (1) above can be simplified. Noting that t|m with m =
t(q − 1) we see that k satisfies (1) iff k satisfies
kn ≡ i − 1

(mod q − 1).

(26)

Moreover, if k is a solution for (1) then k + (q − 1) is as well. Thus one solution
to (26) gives t solutions to (1).

Lemma 15 Let (q, n) be a Dickson pair. For each i, the set {k | (i, k) ∈ T (q, n)}
is either empty or of order gcd(n, q − 1).
Proof: Suppose K = {k|(i, k) ∈ T (q, n)} is nonempty. Let n = n∗ gcd(n, q −
1), q − 1 = q ∗ gcd(n, q − 1) so gcd(n∗ , q ∗ ) = 1. Then k ∈ K implies that
q−1
q−1
(k + gcd(n,q−1)
)n = kn + gcd(n,q−1)
n = kn + (q − 1)n∗ ≡ kn (mod q − 1) so
k+

q−1
gcd(n,q−1)

q−1
∈ K. Thus there is an element of K in {0, . . . , gcd(n,q−1)
− 1}.

q−1
− 1} ∩ K, k1 > k2 .
Suppose there are two. Let k1 , k2 ∈ {0, . . . , gcd(n,q−1)
Then

(k1 − k2 )n ≡ (i − 1) − (i − 1) = 0 (mod q − 1)
⇒ (q − 1)|(k1 − k2 )n

(27)
(28)

⇒ (k1 − k2 )n = c(q − 1) for some c ≥ 1

(29)

⇒ (k1 − k2 )n∗ = cq ∗
k1 , k2 < q ∗ so (k1 − k2 ) < q ∗ ⇒ n∗ > c

(30)
(31)

But n∗ |c by coprimeness of n∗ and q ∗ , which is a contradiction. Thus we cannot
have two distinct k1 , k2 < q ∗ in K. Thus if K is nonempty, it has precisely
gcd(n, q − 1) elements. Writing K as a subset of Zq−1 , K is a coset of hq ∗ i ≤
Zq−1 .

Thus we can calculate the order of our multiplicative automorphism group.
9

Corollary 16 Let (q, n) be a Dickson pair. Then
|Aut(DF (q, n), ∗)| =

1
ϕ(m)t gcd(n, q − 1)
ρ

(32)

where ρ1 is the proportion of i coprime to m for which {k|(i, k) ∈ T (q, n)} is
nonempty.
Proof: Let A = {i|(i, k) ∈ S(q, n)}. This is a subgroup of U(Zm ) of index ρ.
Then by the above two results,
|Aut(Gq,n )| = |A|t gcd(n, q − 1)
1
= ϕ(m)t gcd(n, q − 1).
ρ

(33)
(34)


We can find an explicit description of the factor ρ.
Theorem 17 Let (q, n) be a Dickson pair, q2 the factor of q − 1 containing all
.
the primes in n. Then ρ = ϕ(q2 ) gcd(n,q−1)
q2
Proof: First we show that there exists a k such that (i, k) ∈ S(q, n) iff gcd(i, m) =
1 and gcd(n, q−1)|(i−1). We know that there exists a k such that (i, k) ∈ S(q, n)
iff gcd(i, m) = 1 and kn ≡ i − 1 (mod (q − 1)), so we only need to show the
equivalence of the second requirements. Suppose kn ≡ i − 1 (mod (q − 1)), i.e.
kn − (i − 1) ≡ 0 (mod (q − 1)). Let g = gcd(n, q − 1), n = n̄g and q − 1 = q̄g.
Then
q − 1|kn − (i − 1) ⇔ q̄g|kn̄g − (i − 1)
⇒ g|(i − 1).

(35)
(36)

The other direction requires us to show that divisibility suffices. Let g|i − 1,
so īg = i − 1 for some integer ī. Then kn̄g − īg ≡ 0 (mod q̄g) ⇔ kn̄ − ī ≡ 0
(mod q̄). We know that n̄ is coprime to q̄ so n̄ is a unit in Zq̄ . Thus this equation
is solvable with k = n̄ī and (i, k) ∈ S(q, n), so we have the first part of the proof.
Let A = {i ∈ Zm |there exists a k such that (i, k) ∈ S(q, n) with S(q, n) =
{i ∈ Zm | gcd(i, m) = 1, g|(i − 1)}. We want to know the size of A.
There are two cases to consider, depending upon the value of gcd(n, t).
Case gcd(n, t) = gcd(q − 1, t) = 1. Let q − 1 = q1 q2 where all the primes in
n are in q2 , so gcd(q1 , n) = 1. We know m = t(q − 1) = (tq1 )q2 with tq1 coprime
to q2 . Thus we can write A = {i ∈ Zm | gcd(i, tq1 ) = 1, gcd(i, q2 ) = 1, g|i − 1}.
For the third condition, we can write i = αg + 1 for some α. Because the prime
factors of n and q2 are identical, i is then coprime to q2 . Thus we can write
A = {αg + 1| gcd(αg + 1, tq1 ) = 1, α ∈ {0, . . . ,

(37)

m
− 1}}
g

(38)

q2
− 1}g + 1
g

(39)

Ā = {αg + 1|α ∈ {0, . . . ,
= q2 Zm + {0, . . . ,

m
− 1}}
g

10

when we write sums of sets as the set of all sums. For each β ∈ {0, . . . , qg2 − 1},
let Aβ = q2 Zm + βg + 1. Note that all Aβ are pairwise distinct and that
|Aβ | = |q2 Zm | = tq1 . Then
|{i ∈ Aβ | gcd(i, tq1 ) = 1}| = ϕ(tq1 ).
Thus |A| =

q2
g ϕ(tq1 )

(40)

so
ϕ(m)
|A|
ϕ(tq1 )ϕ(q2 )g
=
q2 ϕ(tq1 )
ϕ(q2 )g
.
=
q2

ρ=

(41)
(42)
(43)

Case gcd(n, t) = 2. The proof is analagous to the proof above, but with
changes for the extra factors of 2. By Theorem 2 we know that n is even and
q ≡ 3 (mod 4). Thus 4 does not divide q − 1, so 4 does not divide n. Let
q − 1 = q1 q2 where all the primes in n are in q2 , so gcd(q1 , n) = 1. We know
m = t(q − 1) = (t̄q1 )(2τ q2 ) with t̄q1 coprime to 2τ q2 , t = 2τ t̄. Thus we can write
A = {i ∈ Zm | gcd(i, tq1 ) = 1, gcd(i, 2τ q2 ) = 1, g|i − 1}. For the third condition,
we can write i = αg + 1. Because the prime factors of n and q2 are identical, i
is then coprime to 2τ q2 . Thus we can write
m
− 1}}
(44)
A = {αg + 1| gcd(αg + 1, t̄q1 ) = 1, α ∈ {0, . . . ,
g
m
− 1}}
(45)
Ā = {αg + 1|α ∈ {0, . . . ,
g
2τ q2
− 1}g + 1
(46)
= (2τ q2 )Zm + {0, . . . ,
g
τ

For each β ∈ {0, . . . , 2 gq2 − 1}, let Aβ = 2τ q2 Zm + βg + 1. Note that all Aβ are
pairwise distinct and that |Aβ | = |2τ q2 Zm | = t̄q1 . Then
|{i ∈ Aβ | gcd(i, t̄q1 ) = 1}| = ϕ(t̄q1 ).
Thus |A| =

2τ q2
g ϕ(t̄q1 )

(47)

so
ρ=
=
=
=
=
=
=

ϕ(m)
|A|
ϕ(t̄q1 )ϕ(2τ q2 )g
2τ q2 ϕ(t̄q1 )
ϕ(2τ q2 )g
2τ q2
ϕ(2τ +1 )ϕ(q2 /2)g
2τ q2
2τ ϕ(q2 /2)g
2τ q2
ϕ(q2 /2)g
q2
ϕ(q2 )g
q2
11

(48)
(49)
(50)
(51)
(52)
(53)
(54)

because s odd implies that ϕ(s) = ϕ(2s).
Thus we are done.

By combining the expressions above, we obtain the following explicit counting result.
Corollary 18 The size of the automorphism group of Gq,n is
tϕ(t)ϕ(q1 )q2
where t =

5.4

qn −1
n(q−1)

(55)

and q − 1 = q1 q2 with all primes factors of n in q2 .

Applying counting structure

While the above results give us an explicit expression for a given Dickson pair,
this is hard to calculate in general. However; we can show that for certain types
of Dickson pairs, there are simpler expressions for the size of the automorphism
group and the index of the nearfield automorphism group in it.
Lemma 19 Let (q, 2) be a Dickson pair, q = pl . Then |S(q, 2)| = ϕ( q
1), so F (DF (q, 2)) =

(q+1)
q2 −1
2l ϕ( 2 ).

2

−1
2 )(q

+

Proof: Following the notation above, q2 = 2τ for some τ . By Theorem 17
τ −1
we have that ρ = ϕ(q2 ) gcd(n,q−1)
. Now using
= 2 2τ gcd(2, q − 1) = gcd(2,q−1)
q2
2
Corollary 16 we get that
1
ϕ(m)t gcd(2, q − 1)
ρ
2
q2 − 1 
=
ϕ
t gcd(2, q − 1)
gcd(2, q − 1)
2

Aut(DF (q, n), ∗)| =

(56)
(57)
(58)

2

q − 1
=ϕ
(q + 1)
2

(59)

As we saw above, the order of the nearfield automorphism group is ln/k, where
k is the order of p modulo n. But n = 2 and p is odd, so k = 1. So the nearfield
automorphism group has order 2l. Thus our factor is the order of S(q, 2) divided
by 2l.

Similar results can be obtained for n = 3 and others.

6

Summary

We have investigated the structure of the multiplicative group of a finite nearfield
and found explicit counts for the Dickson nearfield case, as well as the seven
exceptional finite nearfields.
These results can be used to count the number of nonisomorphic near vector
spaces of a given dimension over a given finite nearfield. We showed that finite
near vector spaces have isomorphism classes that can be readily defined by
sequences of cosets. Thus the process of enumerating examples is based upon
questions of the multiplicative and nearfield automorphism groups and the index
of the latter in the former.
12

The main open problems remaining here relate to the explicit construction
of the automorphism groups. Numerically we were able to find the size of the
automorphism group in all cases, using some number theoretical properties. We
were able to show that in the case that gcd(n, t) = 1, the multiplicative group
of a Dickson nearfield is a split metacyclic group and thus we can in principle
determine the automorphism group. However we were unable to find an explicit
expression for the automorphism group. It would appear that techniques similar
to those used in Theorem 17 could be applied to determine the automorphism
group explicitly.
Some smaller questions require attention. Can we explicitly describe the
inner automorphisms? Does this tell us something important about the automorphism group?
These results show that there are a large number of nonisomorphic near
vector spaces of a given dimension over a given nearfield. Some applications of
near vector spaces have been proposed [2] and this quantity of examples offers
some indication of their rich structure. In [3] one author and Gerhard Wendt
have shown that near vector space type constructions, in particular homeomorphisms of near vector spaces, can be used to construct nearrings with special
properties, generalising the properties of units acting as endomorphisms on the
additive group of a nearfield. This is an enticing direction of further research.

Acknowledgements
We would like to thank Peter Mayr and the referee for some very useful suggestions. This work has been partially supported by Austrian Science Funding
(FWF) Project number 23689-N18 and by the South African National Research
Foundation.

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14

