Groups generated by derangements

Abstract We examine the subgroup D ( G ) of a transitive permutation group G which is generated by the derangements in G. Our main results bound the index of this subgroup: we conjecture that, if G has degree n and is not a Frobenius group, then | G : D ( G ) | ⩽ n − 1 ; we prove this except when G is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding | H : R ( H ) | , where H is a linear group on a finite vector space and R ( H ) is the subgroup of H generated by elements having eigenvalue 1. If G is a Frobenius group, then D ( G ) is the Frobenius kernel, and so G / D ( G ) is isomorphic to a Frobenius complement. We give some examples where D ( G ) ≠ G , and examine the group-theoretic structure of G / D ( G ) ; in particular, we construct groups G in which G / D ( G ) is not a Frobenius complement.