Let G be a permutation group on a finite set Ω.
A sequence B=(ω1, …, ωb) of
points in Ω is called a base if its pointwise stabilizer in G
is the identity. Bases are of
fundamental importance in computational algorithms for permutation groups. For
both practical and theoretical reasons, one is interested in the minimal base size for
(G, Ω), For a nonredundant base B, the elementary inequality
2[mid ]B[mid ][les ][mid ]G[mid ][les ][mid ]Ω[mid ][mid ]B[mid ]
holds; in particular,
[mid ]B[mid ][ges ]log[mid ]G[mid ]/log[mid ]Ω[mid ].
In the case when G is primitive on Ω, Pyber
[8, p. 207] has conjectured that the minimal base size is less than
Clog[mid ]G[mid ]/log[mid ]Ω[mid ] for
some (large) universal constant C.
It appears that the hardest case of Pyber's conjecture is that of primitive affine
groups. Let H=GV be a primitive affine group;
here the point stabilizer G acts
faithfully and irreducibly on the elementary abelian regular normal subgroup V of
H, and we may assume that Ω=V. For positive integers
m, let mV denote the direct sum of m copies of V. If
(v1, …, vm)∈mV
belongs to a regular G-orbit, then
(0, v1, …, vm) is a base for the
primitive affine group H. Conversely, a base
(ω1, …, ωb) for H
which contains 0∈V=Ω gives rise to a regular
G-orbit on (b−1) V. Thus Pyber's
conjecture for affine groups can be viewed as a regular orbit problem for G-modules,
and it is therefore a special case of an important problem in group representation
theory. For a related result on regular orbits for quasisimple groups,
see [4, Theorem 6].