Nearfield Properties#

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

•: NF1: \((N,+)\) is an abelian group and \(0\) is its identity element. Let \(N^\times\) denote the set of non-zero elements of \(N\).
•: NF2: \((N^\times,\circ)\) is a group and \(1\) is its identity element.
•: NF3: \(a\circ 0 = 0\circ a = 0\) for all \(a\in N\).
•: NF4: \((a+b)\circ c = a\circ c + b\circ c\) for all \(a,b,c \in N\).

A subset \(S\) of a nearfield \(N\) is a sub-nearfield if \((S,+)\) and \((S\setminus \{0\},\circ)\) are groups. The sub-nearfield generated by a subset \(X\) is the intersection of all sub-nearfields containing \(X\). The prime field \({\cal P}(N)\) of \(N\) is the sub-nearfield generated by \(1\).

The inverse of \(x\in N^\times\) is written \(x^{[-1]}\). But where no confusion is possible we write multiplication of nearfield elements \(x\) and \(y\) as \(xy\) rather than \(x\circ y\) 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 \({\bf F}_{p}\) 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 \(p^n\) for some \(n\). Every field is a nearfield.

If \(N\) is a nearfield, the centre of \(N\) is the set

\[{\cal Z}(N) = \{\, x \in N \mid xy = yx\text{ for all }y\in N\,\}\]

and the kernel of \(N\) is the subfield

\[{\cal K}(N) = \{\, x \in N \mid x(y+z) = xy + xz\text{ for all }y,z\in N\,\}.\]

It is clear that \({\cal Z}(N) \subseteq {\cal K}(N)\) but equality need not hold because, in general, \({\cal Z}(N)\) need not be closed under addition. Furthermore, the prime field \({\cal P}(N)\) need not be contained in \({\cal Z}(N)\). However, for the Dickson nearfields \({\cal Z}(N) = {\cal K}(N)\).

If \(N\) is a nearfield, then \({\cal Z}(N) = \bigcap\{ {\cal K}(N)^x \mid x \in N, x \ne 0\}\).

Sharply Doubly Transitive Groups#

A group \(G\) acting on a set \(\Omega\) is sharply doubly transitive if \(G\) is doubly transitive on \(\Omega\) and only the identity element fixes two points.

If \(G\) is a finite sharply doubly transitive group on \(\Omega\) then

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 \(p^n\) for some \(n\) and some prime \(p\).
Addition and multiplication between elements of \(\Omega\) can be defined so that \(\Omega\) becomes a nearfield and so that the group \(G\) is isomorphic to the group of all affine transformations \(v \mapsto va+b\) of \(\Omega\), where \(a\in \Omega^\times\) and \(b\in \Omega\).

There is a converse to this theorem, namely if \(N\) is a nearfield, the group of all transformations \(v \mapsto va+b\) acts sharply doubly transitively on \(N\).

Let \(F\) be the prime field of \(N\), regard \(N\) as a vector space over \(F\) and define \(\mu : N^\times \to {\operatorname{GL}}(N)\) by \(v^{\mu(a)} = va\). Then for all \(a\in N^\times\), \(a \ne 1\), the linear transformation \(\mu(a)\) is fixed-point-free. Furthermore, \(\mu\) defines an isomorphism between the multiplicative group \(N^\times\) and its image in \({\operatorname{GL}}(N)\).

Suppose that \(G = H\ltimes M\) is a sharply doubly transitive group of degree \(p^n\), as above. The centre of \(G\) is trivial and \(M\) is a minimal normal subgroup. Thus if \(\Omega'\) is a minimal permutation representation we may suppose that it is primitive. Then \(M\) is transitive on \(\Omega'\) and since \(M\) is abelian, it acts regularly on \(\Omega'\). Thus \(p^n\) is the minimal degree of a faithful permutation representation of \(G\).