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A (right-distributive) nearfield is a set N containing elements
0 and 1 and with binary operations + and  such that
- NF1: (N, + ) is an abelian group and 0 is its identity element.
Let N x denote the set of non-zero elements of N.
- NF2: (N x , ) is a group and 1 is its identity element.
- NF3: a 0 = 0 a = 0 for all a∈N.
- NF4: (a + b) c = a c + b c for all a, b, c ∈N.
A subset S of a nearfield N is a sub-nearfield if (S, + ) and
(S - {0}, ) are groups. The sub-nearfield generated
by a subset X is the intersection of all sub-nearfields containing X.
The prime field (P)(N) of N is the sub-nearfield generated
by 1.
The inverse of x∈N x is written x[ - 1]. But where no confusion
is possible we write multiplication of nearfield elements x and y as
xy rather than x y and we write the inverse of x as x - 1. (In
the Magma code we use "*" as the symbol for multiplication.)
If N is a finite nearfield, the prime field of N is a Galois field
GF(p) for some prime p and p is the characteristic of N.
A nearfield of characteristic p is a vector space over its
prime field and therefore its cardinality is pn for some n.
Every field is a nearfield.
If N is a nearfield, the centre of N is the set
(Z)(N) = { x ∈N | xy = yx for all y∈N }
and the kernel of N is the subfield
(K)(N) = { x ∈N | x(y + z) = xy + xz for all
y, z∈N }.
It is clear that (Z)(N) ⊆(K)(N) but equality need not
hold because, in general, (Z)(N) need not be closed under addition.
Furthermore, the prime field (P)(N) need not be contained in
(Z)(N). However, for the Dickson nearfields (Z)(N) = (K)(N).
If N is a nearfield, then (Z)(N) = bigcap{ (K)(N)x | x ∈N, x ≠0}.
A group G acting on a set Ω is sharply doubly transitive if
G is doubly transitive on Ω and only the identity element fixes
two points.
If G is a finite sharply doubly transitive group on Ω then
- 1.
- The set M consisting of the identity element and the elements of G
without fixed points is an elementary abelian normal subgroup of G of
order pn for some n and some prime p.
- 2.
- Addition and multiplication between elements of Ω can be
defined so that Ω becomes a nearfield and so that the
group G is isomorphic to the group of all affine transformations
v |-> va + b of Ω, where a∈Ω x and b∈Ω.
There is a converse to this theorem, namely if N is a nearfield, the group
of all transformations v |-> va + b acts sharply doubly transitively
on N.
Let F be the prime field of N, regard N as a vector space over F
and define μ : N x to GL(N) by
vμ(a) = va. Then for all a∈N x , a ≠1, the linear
transformation μ(a) is fixed-point-free.
Furthermore, μ defines an isomorphism between the multiplicative group
N x and its image in GL(N).
Suppose that G = H ltimes M is a sharply doubly transitive group of degree
pn, as above. The centre of G is trivial and M is a minimal normal
subgroup. Thus if Ω' is a minimal permutation representation we may
suppose that it is primitive. Then M is transitive on Ω' and since
M is abelian, it acts regularly on Ω'. Thus pn is the minimal
degree of a faithful permutation representation of G.
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