Primitive permutation IBIS groups

Abstract Let G be a finite permutation group on Ω. An ordered sequence of elements of Ω, ( ω 1 , … , ω t ) , is an irredundant base for G if the pointwise stabilizer G ( ω 1 , … , ω 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 ) × PSL ( 2 , 2 f ) for some f ≥ 2 , in its diagonal action of degree 2 f ( 2 2 f − 1 ) .