A reduction theorem for nonsolvable finite groups

Every finite group G has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the nonsolvable length of the group and denoted by λ(G). For every integer n, we define a particular class of groups of nonsolvable length n, called n-rarefied, and we show that every finite group of nonsolvable length n contains an n-rarefied subgroup. As applications of this result, we improve the known upper bounds on λ(G) and determine the maximum possible nonsolvable length for permutation groups and linear groups of fixed degree resp. dimension.