##
**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 635.10040

**Autor: ** Erdös, Paul; Mays, Michael E.

**Title: ** On nilpotent but not abelian groups and abelian but not cyclic groups. (In English)

**Source: ** J. Number Theory 28, No.3, 363-368 (1988).

**Review: ** Using general sieve-type methods of number theory and certain density estimates for prime numbers, the authors derive asymptotic formulae for A(n)-C(n) and N(n)-A(n), where A(n) = \#**{**m \leq n: every group of order m is abelian**}**, C(n) = \#**{**m \leq n: every group of order m is cyclic**}**, and N(n) = \#**{**m \leq n: every group of order m is nilpotent**}**.

The second author [Arch. Math. 31, 536-538 (1978; Zbl 388.20021)] and *E. J. Scourfield* [Acta Arith. 29, 401-423 (1976; Zbl 286.10023)] showed previously that asymptotically all three of the above counting functions have the form (1+o(1))ne^{-\gamma}/ log_{3}n.

The present authors now prove that there exist constants c_{1},c_{2} such that A(n)-C(n) = (1+o(1))c_{1}n/(log_{2}n)(log_{3}n)^{2},

N(n)-A(n) = (1+o(1))c_{2}n/(log_{2}n)^{2}(log_{3}n)^{2}.

**Reviewer: ** J.Knopfmacher

**Classif.: ** * 11N45 Asymptotic results on counting functions for other structures

20K99 Abelian groups

20D99 Abstract finite groups

**Keywords: ** asymptotic formulae

**Citations: ** Zbl 396.20018; Zbl 388.20021; Zbl 286.10023

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag