Let $\pi $ be a set of primes. We say that a group G satisfies $D_{\pi }$ if G possesses a Hall $\pi $ -subgroup H and every $\pi $ -subgroup of G is contained in a conjugate of H. We give a new $D_{\pi }$ -criterion following Wielandt’s idea, which is a generalisation of Wielandt’s and Rusakov’s results.

The Wielandt subgroup of a group $G$, denoted by ${\it\omega}(G)$, is the intersection of the normalisers of all subnormal subgroups of $G$. The terms of the Wielandt series of $G$ are defined, inductively, by putting ${\it\omega}_{0}(G)=1$ and ${\it\omega}_{i+1}(G)/{\it\omega}_{i}(G)={\it\omega}(G/{\it\omega}_{i}(G))$. In this paper, we investigate the relations between the$p$-length of a $p$-soluble finite group and the Wielandt series of its Sylow $p$-subgroups. Some recent results are improved.