An upper bound for the nonsolvable length of a finite group in terms of its shortest law

Abstract

Every finite group G $G$ 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 $G$ , is called the nonsolvable length λ ( G ) $\lambda (G)$ of G $G$ . 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 $n$ , no nontrivial word of length at most n $n$ (in any number of variables) can be a law. This result is then used to give a bound on λ ( G ) $\lambda (G)$ in terms of the length of the shortest law of G $G$ , thus confirming a conjecture of Larsen. Moreover, we give a positive answer to a problem raised by Khukhro and Larsen concerning the non-2-solvable length of finite groups.