Primitive permutation IBIS groups

Abstract

Let $G$ be a finite permutation group on $\Omega$. An ordered sequence of elements of $\Omega$, $({\omega }_1,\dots ,{\omega }_t)$, is an irredundant base for $G$ if the pointwise stabilizer $G_{({\omega }_1,\dots ,{\omega }_t)}$ is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same size we say that $G$ is an IBIS group. In this paper we show that if a primitive permutation group is IBIS, then it must be almost simple, of affine-type, or of diagonal type. Moreover we prove that a diagonal-type primitive permutation groups is IBIS if and only if it is isomorphic to $PSL(2,2^f)\times PSL(2,2^f)$ for some $f\ge 2,$ in its diagonal action of degree $2^f(2^{2f}-1).$