Primitive permutation IBIS groups

TL;DR

This paper characterizes primitive IBIS groups, showing they are either almost simple, affine, or diagonal type, and precisely identifies diagonal IBIS groups as certain products of PSL(2,2^f).

Contribution

It classifies primitive IBIS groups and provides a complete characterization of diagonal-type IBIS groups.

Findings

Primitive IBIS groups are almost simple, affine, or diagonal type.

Diagonal IBIS groups are isomorphic to PSL(2,2^f)×PSL(2,2^f).

Diagonal IBIS groups occur in degree 2^f(2^{2f}-1).

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 $f\geq 2,$ in its diagonal action of degree $2^f(2^{2f}-1).$