  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 $|G:D(G)|\leqslant\sqrt{n}-1$; 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
$D(G)\ne G$, 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.
