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On orders solely of abelian groups

Published online by Cambridge University Press:  18 May 2009

S. Srinivasan
S. Srinivasan
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India
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Let n = be the factorization of an integer n(>1) into prime powers, and set Φ(n):= . In particular, for squarefree n, Φ(n) = phi;(n). Consider the set

.

It is known (from [5]) that A consists precisely of those integers n for which there is no non-abelian group of order n. It is also known (from [7]) that the set

consists solely of integers n with the property that every group of order n is cyclic. We set C′ = A – C.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 29 , Issue 1 , January 1987 , pp. 105 - 108
Copyright
Copyright © Glasgow Mathematical Journal Trust 1987

References

REFERENCES

1.Erdös, P., Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.) 12 (1948), 75–78.Google Scholar
2.Hall, R. R., Halving an estimate obtained from Selberg's upper bound method, Acta Arith. 25 (1973/1974), 347–351.CrossRefGoogle Scholar
3.Hardy, G. H. and Wright, E. M., Introduction to the theory of numbers, 4th edn. (Oxford University Press, 1964).Google Scholar
4.Montgomery, H. L., A note on the large sieve, J. London Math. Soc. 43 (1968), 93–98.CrossRefGoogle Scholar
5.Rédei, L., Das “schiefe Produkt” in der Gruppen theorie, Comment. Math. Helv. 20 (1947), 225–264.CrossRefGoogle Scholar
6.Richert, H.-E., Sieve methods (Tata Institute Lecture Notes, Bombay, 1976).Google Scholar
7.Szele, T., Uber die endlichen Ordnungszahlen, zu denen nur eine Gruppe gehort, Comment. Math. Helv. 20 (1947), 265–267.CrossRefGoogle Scholar
8.Warlimont, R., On the set of natural numbers which only yield orders of abelian groups, J. Number Theory 20 (1985), 354–362.CrossRefGoogle Scholar