On the Average Number of Groups of Square-Free Order

Abstract

Let $G(n)$ denote the number of (nonisomorphic) groups of order $n$. It is shown here that for large $x$ \[ x^{1.68} \leq \sum \nolimits ’_{n \leq x} G(n) \leq {x^2} \cdot \exp \{ -(1 + \mathrm {o}(1)) \log x\log \log \log x/\log \log x\} ,\] where $\sum ’$ denotes a sum over square-free $n$. Under an unproved hypothesis on the distribution of primes $p$ with all primes in $p - 1$ small, it is shown that the upper bound is tight.