  Every finite group $G$ has a normal series each of whose factors is either a
solvable group or a direct product of non-abelian simple groups. The minimum
number of nonsolvable factors, attained on all possible such series in $G$, is
called the \emph{nonsolvable length} $\lambda(G)$ of $G$. In the present paper,
we prove a theorem about permutation representations of groups of fixed
nonsolvable length. As a consequence, we show that in a finite group of
nonsolvable length at least $n$, no non-trivial word of length at most $n$ (in
any number of variables) can be a law. This result is then used to give a bound
on $\lambda(G)$ in terms of the length of the shortest law of $G$, thus
confirming a conjecture of Larsen. Moreover our Theorem C can be used to give a
positive answer, in the case $p=2$, to a problem raised by Khukhro and
Shumyatsky, concerning the non-$p$-solvable length of finite groups.
