Numbers which are only orders of abelian or nilpotent groups

Just, Matthew

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 453-466

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

Soient $C\left(x\right)$, $A\left(x\right)$ et $N\left(x\right)$ les fonctions qui comptent le nombre de $n\le x$ tels que chaque groupe d’ordre $n$ soit respectivement cyclique, abélien et nilpotent. En affinant un résultat de Erdős et Mays, on donne des développements asymptotiques des fonctions $A\left(x\right)-C\left(x\right)$ et $N\left(x\right)-A\left(x\right).$

Classification : 11N37, 20D60
Keywords: group numbers, asymptotic series

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