## 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/ log3n.
The present authors now prove that there exist constants c1,c2 such that

A(n)-C(n) = (1+o(1))c1n/(log2n)(log3n)2,

N(n)-A(n) = (1+o(1))c2n/(log2n)2(log3n)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

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