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. The main focus of this paper is to give asymptotic bounds for the isomorphism classes of solvable cube-free groups of order $n$. Our bound allows us to show that there are no more than $n^4$ such groups of order $n$ up to isomorphism, which improves on the existing bounds known. We develop methods that allow us to give a formula for the number of conjugacy classes of subgroups that are maximal amongst non-abelian solvable cube-free $p^{\prime }$-subgroups of ${\rm GL}(2,q)$, where $p$ is a prime and $q = p^k$ for some positive integer $k$. Further, we find the exact number of split extensions of $P$ by $Q$, up to isomorphism of a given order, where $P \in \lbrace {\mathbb {Z}}_p \times {\mathbb {Z}}_p, {\mathbb {Z}}_{p^{\alpha }}\rbrace$, $p$ is a prime, $\alpha$ is a positive integer and $Q$ is a cube-free abelian group of odd order such that $p \nmid |Q|$. These results are used in proving the upper bound and in finding a lower bound for the isomorphism classes of solvable cube-free groups of order $n$.