# 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

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 $p^n$ for some $n$ and some prime $p$.

2. 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 $G$.