  A finite group $G$ is called a Schur group if every $S$-ring over $G$ is
schurian, i.e. associated in a natural way with a subgroup of $Sym(G)$ that
contains all right translations. One of the crucial open questions in the
$S$-ring theory is the question on existence of an infinite family of
nonabelian Schur groups. We prove that a dihedral group of order $2p$, where
$p$ is a Fermat prime or prime of the form $p=4q+1$, where $q$ is also prime,
is Schur. Towards this result, we prove nonexistence of a difference set in a
cyclic group of order $p\neq 13$ and classify all $S$-rings over some dihedral
groups.
