Numbers which are only orders of abelian or nilpotent groups

TL;DR

This paper refines asymptotic formulas for counting integers up to x where all groups of that order are abelian but not cyclic, or nilpotent but not abelian, extending previous results by Erdos and Mays.

Contribution

It provides detailed asymptotic series expansions for these counting functions, improving understanding of the distribution of such group orders.

Findings

Asymptotic series expansions for A(x)-C(x) and N(x)-A(x) are derived.

The results refine previous bounds by Erdos and Mays.

New insights into the distribution of group orders with specific properties.

Abstract

Refining a result of Erdos and Mays, we give asymptotic series expansions for the functions $A(x)-C(x)$, the count of $n\leq x$ for which every group of order $n$ is abelian (but not all cyclic), and $N(x)-A(x)$, the count of $n\leq x$ for which every group of order $n$ is nilpotent (but not all abelian).