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

Abstract

A group is said to be cube-free if its order is not divisible by the cube of any prime. Let $f_{cf,sol}(n)$ denote the isomorphism classes of solvable cube-free groups of order $n$. We find asymptotic bounds for $f_{cf,sol}(n)$ in this paper. Let $p$ be a prime and let $q = p^k$ 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 ${\rm GL}(2,q)$. Further, we find the exact number of split extensions of $P$ by $Q$ up to isomorphism of a given order where $P \in \{{\mathbb Z}_p \times {\mathbb Z}_p, {\mathbb Z}_{p^{\alpha}}\}$, $p$ is a prime, $\alpha$ is a positive integer and $Q$ is a cube-free abelian group of odd order such that $p \nmid |Q|$.