15.66 (2002)

For a class $\mathfrak{X}$ of groups let $g_\mathfrak{X}(n)$ be the number of groups of order $n$ in the class $\mathfrak{X}$ (up to isomorphism). Many years ago I formulated the following problem: find good upper bounds for the quotient $g_\mathfrak{V}(n)/g_\mathfrak{U}(n)$, where $\mathfrak{V}$ is a variety that is defined by its finite groups and $\mathfrak{U}$ is a subvariety of $\mathfrak{V}$. (This quotient is not defined for all $n$ but only for those for which there are groups of order $n$ in $\mathfrak{U}$.) Some progress has been made by G. Venkataraman (Quart. J. Math. Oxford (2), 48, no. 189 (1997), 107–125) when $\mathfrak{V}$ is a variety generated by finite groups all of whose Sylow subgroups are abelian. Conjecture: if $\mathfrak{V}$ is a locally finite variety of $p$-groups and $\mathfrak{U}$ is a non-abelian subvariety of $\mathfrak{V}$, then $g_\mathfrak{V}(p^m)/g_\mathfrak{U}(p^m) < p^{O(m^2)}$.

Moreover, this seems a possible way to attack the Sims Conjecture that when we write the number of groups of order $p^m$ as $p^{\frac{2}{27} m^3 + \varepsilon(m)}$ the error term $\varepsilon(m)$ is $O(m^2)$.