Enumerating finite groups without abelian composition factors

Let Σ denote the class of allnon-abelian finite simple groups. We are concerned with enumerating poly-Σ groups, that is finite groups without abelian composition factors. For any natural numbern let gΣ(n) denote the number of (isomorphism classes of) poly-Σ groups of order at mostn. We determine the growth rate of the sequence gΣ(n),n ε ℵ. Similarly, for anyS ε Σ we give estimates for the numbers ĝS(k) of poly-S groups of composition length at mostk, ask tends to infinity. This initiates an investigation somewhat complementary to the “classical” enumeration of finitep-groups by Higman 6] and Sims 15]. Our ancillary results include upper bounds for the minimal number of generators and for the number of (equivalence classes of) permutation actions of any given poly-Σ group.