Numbers which are only orders of abelian or nilpotent groups

Matthew Just

doi:10.5802/jtnb.1251

Matthew Just ¹

¹ Department of Mathematics University of Georgia Athens GA 30602 United States

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

Résumé
Abstract

Soient $C (x)$ , $A (x)$ et $N (x)$ les fonctions qui comptent le nombre de $n \leq x$ 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 $A (x) - C (x)$ et $N (x) - A (x) .$

Refining a result of Erdős 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).

Reçu le : 2021-10-04
Révisé le : 2023-01-03
Accepté le : 2023-01-26
Publié le : 2023-10-10

DOI : 10.5802/jtnb.1251

Classification : 11N37, 20D60
Mots clés : group numbers, asymptotic series

Affiliations des auteurs :

Matthew Just ¹

¹ Department of Mathematics University of Georgia Athens GA 30602 United States

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2023__35_2_453_0,
     author = {Matthew Just},
     title = {Numbers which are only orders of abelian or nilpotent groups},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {453--466},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {35},
     number = {2},
     year = {2023},
     doi = {10.5802/jtnb.1251},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1251/}
}

TY  - JOUR
AU  - Matthew Just
TI  - Numbers which are only orders of abelian or nilpotent groups
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2023
SP  - 453
EP  - 466
VL  - 35
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1251/
DO  - 10.5802/jtnb.1251
LA  - en
ID  - JTNB_2023__35_2_453_0
ER  -

%0 Journal Article
%A Matthew Just
%T Numbers which are only orders of abelian or nilpotent groups
%J Journal de théorie des nombres de Bordeaux
%D 2023
%P 453-466
%V 35
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1251/
%R 10.5802/jtnb.1251
%G en
%F JTNB_2023__35_2_453_0

Matthew Just. Numbers which are only orders of abelian or nilpotent groups. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 453-466. doi : 10.5802/jtnb.1251. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1251/

Bibliographie
Cité par

[1] Aleksandr V. Begunts On an asymptotic formula for $F_{1} (x)$ , Vestn. Mosk. Univ., Volume 5 (2001), pp. 57-59

[2] Leonard E. Dickson Definitions of a group and a field by independent postulates, Trans. Am. Math. Soc., Volume 6 (1905), pp. 198-204 | DOI | MR | Zbl

[3] Pál Erdős Some asymptotic formulas in number theory, J. Indian Math. Soc., Volume 12 (1948), pp. 75-78 | MR | Zbl

[4] Pál Erdős; Michael E. Mays On nilpotent but not abelian groups and abelian but not cyclic groups, J. Number Theory, Volume 28 (1988) no. 3, pp. 363-368 | DOI | MR | Zbl

[5] Heini Halberstam; Hans-Egon Richert Sieve methods, London Mathematical Society Monographs, 4, Academic Press Inc., 1974

[6] Michael E. Mays Counting abelian, nilpotent, and supersolvable group orders, Arch. Math., Volume 31 (1978), pp. 536-538 | DOI | MR

[7] M. J. Narlikar; S. Srinivasan On orders solely of abelian groups. II, Bull. Lond. Math. Soc., Volume 20 (1988) no. 3, pp. 211-216 | DOI | MR | Zbl

[8] Karl K. Norton On the number of restricted prime factors of an integer I, Ill. J. Math., Volume 20 (1976), pp. 681-705 | MR | Zbl

[9] Gerhard Pazderski Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehören, Arch. Math., Volume 10 (1959), pp. 331-343 | DOI | MR | Zbl

[10] Paul Pollack Numbers which are orders only of cyclic groups (submitted)

[11] Carl Pomerance On the distribution of amicable numbers, J. Reine Angew. Math., Volume 293/294 (1977), pp. 217-222 | MR | Zbl

[12] Eira J. Scourfield An asymptotic formula for the property $(n, f (n)) = 1$ for a class of multiplicative functions, Acta Arith., Volume 29 (1976) no. 4, pp. 401-423 | DOI | MR

[13] Tibor Szele Über die endlichen Ordnungszahlen, zu denen nur eine Gruppe gehört, Comment. Math. Helv., Volume 20 (1947), pp. 265-267 | DOI | Zbl

[14] Edward C. Titchmarsh A divisor problem, Rend. Circ. Mat. Palermo, Volume 54 (1930), pp. 414-429 | DOI | Zbl

Cité par Sources :