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Nearfield Properties

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}.

Subsections

Sharply Doubly Transitive Groups

A group G acting on a set Ωis sharply doubly transitive if G is doubly transitive on Ωand only the identity element fixes two points.

If G is a finite sharply doubly transitive group on Ωthen

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 Ωcan be defined so that Ωbecomes 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, μdefines 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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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012