Enumeration of solvable cube-free groups and counting certain types of split extensions

A group is said to be cube-free if its order is not divisible by the cube of any prime. Let fcf,sol(n) denote the isomorphism classes of solvable cube-free groups of order n. We find asymptotic bounds for fcf,sol(n) in this paper. Let p be a prime and let q = pᵏ for some positive integer k. We also give a formula for the number of conjugacy classes of the subgroups that are maximal amongst non-abelian solvable cube-free $p'$-subgroups of GL(2,q). Further, we find the exact number of split extensions of P by Q up to isomorphism of a given order where P ∈ \ Zₚ × Zₚ, Zp^α\, p is a prime, $α$ is a positive integer and Q is a cube-free abelian group of odd order such that p |Q|.