A (right-distributive) nearfield is a set N containing elements
0 and 1 and with binary operations + and such that
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 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
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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