ADDITION.
On p. 229 in Section 7.7, before `Problem' insert: `The case
where $G$ is almost simple of Lie type has been treated in
.SC Cuypers
.[[
Cuypers thesis 1989
.]]'.
ADDITION.
On p. 230 in Section 7.7, before `Problem' insert:
.LP
The case of the Lyons group has been treated in
.SC Soicher
.[[
Soicher Lyons 1993
.]].
.LP
In case (iii) significant progress has been made by
.SC van Bon
.[[
van Bon Thesis Utrecht 1990
.]]:
.LP
$Th$.
.I
Let G be as described in (iii) of Theorem 7.7.1. Suppose
that \(*G has diameter $X> 2$ and valency $X>= 3$. Write
$V GAMMA = Ff sub p sup m$ so that $G sub 0$ is the
stabilizer of the zero vector and embeds in $GL( m,p)$.
Take $q vb p sup m$ maximal such that
the vector space $Ff sub p sup m$
carries an $Ff sub q$-structure, denoted by $V$, preserved by $G sub 0$
(that is, $G sub 0 <= GAMMA L (V)$).
Suppose further that $GAMMA$ is neither a bilinear forms graph
nor a Hamming graph.
Then
either $V = Ff sub q$, $q = p sup m$ and
$G sub 0$ is conjugate to a subgroup of $GAM L sub 1 (q )$
or
the generalized
Fitting subgroup $F sup \(** (G sub 0 "/" Z (G sub 0 ))$ is a simple
group whose projective representation on V is
absolutely irreducible and can be realized over no proper subfield of
$GF(q)$.
.LP
This theorem heavily depends on
.SC Aschbacher
.[[
Aschbacher 1984 maximal subgroups
.]].
.LP
Assuming the classification of finite simple
groups, the determination of affine distance-transitive graphs
whose point-stabilizer is almost simple (and nonabelian) is
a matter of diligently ploughing through various possibilities.
It is to be expected that this will be finished soon.
(The case where $G sub 0$ leaves an orthogonal or unitary form
invariant is treated in
.SC van Bon
.[[
van Bon affine with quadratic forms
.]].)
Thus, the following remains:
.sp
.B
Problem.
.R
Classify all distance-transitive graphs (or even groups)
whose automorphism groups contain the additive group
$A$ of the 1-dimensional $Ff sub q$-space $V$
and is contained in $A GAMMA L(V) isom A GAMMA L(1,q)$.
Those with diameter 2 have been classified: they are
the Paley graphs, cf.
.SC Liebeck
.[[
Liebeck affine rank three 1987
.]].
ADDITION.
On p. 231, at the end of \(sc7.7, add: `An extensive survey
on the topic of this section is given by
.SC Ivanov
.[[
Ivanov classification distance transitive graphs their preprint
.]].'
ADDITION.
On p. 232, at the end of the section on distance-transitive
digraphs, add:
.SC `Leonard & Nomura
.[[
Leonard Nomura girth directed 1993
.]]
showed that every distance-regular digraph of short type,
(that is, with $d = g-1$) has $d <= 7$.'
ADDITION.
On p. 232, line $-18$, insert: `(see also
.SC Cameron
.[[
Cameron oligomorphic 1990
.]])'.
ADDITION.
On p. 234, add at the end of the page: `(iii)
Of course the graphs of Lie type, as discussed in Theorem 10.7.2
but for an infinite field, also provide examples
of distance-transitive graphs of infinite valency'.