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On The Number of Groups of Squarefree Order

Published online by Cambridge University Press:  20 November 2018

M. Ram Murty  and
S. Srinivasan
M. Ram Murty
Affiliation:
Department of Mathematics McGill University Montreal, CanadaH3A 2K6
S. Srinivasan
Affiliation:
School of Mathematics Tata Institute of Fundamental Research Bombay 400 005, India
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Abstract

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Let G(n) denote the number of non-isomorphic groups of order n. We prove that for squarefree integers n, there is a constant A such that

where denotes the Euler function. This upper bound is essentially best possible, apart from the constant A.

Keywords

10H26
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 4 , 01 December 1987 , pp. 412 - 420
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Adleman, L.M., Pomerance, C., Rumely, R.S., On distinguishing prime numbers from composite numbers, Annals of Mathematics, 117 (1983), pp. 173–206.Google Scholar
2. Erdös, P., Ram Murty, M., Kumar Murty, V., On the enumeration of finite groups, Journal of Number Theory, 25 (1987), pp. 360–378.Google Scholar
3. Hölder, O., Die gruppen mit quadratfreier ordnungszahl, Nachr. Konigl. Ges. Wiss. Gottingen Math.-Phys. Kl (1895), pp. 211–229.Google Scholar
4. Ram Murty, M., Kumar Murty, V., On the number of groups of a given order, Journal of Number Theory, 18 (1984), pp. 178–191.Google Scholar
5. Ram Murty, M., On groups of squarefree order, Math. Annalen, 267 (1984), pp. 299–309.Google Scholar
6. Neumann, P., An enumeration theorem for finite groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), pp. 395–401.Google Scholar
7. Prachar, K., Uber die anzahl der Teiler einer naturlichen zahl welche die form p — 1 haben, Monatsh. Math. 59 (1955), pp. 91–97.Google Scholar
8. Prachar, K., Primzahlverteilung, Die grundlehren der Math. Wiss, in Einze/darst, Vol. 91, Berlin, Springer, 1957.Google Scholar
9. Pomerance, C., On the average number of groups of squarefree order, (to appear in the Proceedings of the A.M.S.).Google Scholar