zbMATH — the first resource for mathematics

Perfect numbers and finite groups. (English) Zbl 1280.20026
A number is said to be perfect if it is the sum of its proper divisors. In the current paper this notion is extended to finite groups by calling a finite group a Leinster group if its order is equal to the sum of the orders of all its proper normal subgroups. Several general results and examples of Leinster groups are presented.

20D60 Arithmetic and combinatorial problems involving abstract finite groups
11A25 Arithmetic functions; related numbers; inversion formulas
Full Text: DOI
[1] C. W. ANDERSON, The solution of S(n) \? s(n)=n \? a=b, F(n) \? W(n)=n \? a=b and some related considerations, unpublished manuscript (1974).
[2] T. DE MEDTS, Recovering n from s(n)=n, MathOverflow, http://mathover- flow.net/questions/56376.
[3] T. DE MEDTS - M. TAÆRNAÆUCEANU, Finite groups determined by an inequality of the orders of their subgroups, Bull. Belg. Math. Soc. Simon Stevin 15, no. 4 (2008), pp. 699-704. · Zbl 1166.20017 · euclid:bbms/1225893949
[4] B. HUPPERT, Endliche Gruppen I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin, New York, 1967.
[5] T. LEINSTER, Perfect numbers and groups, arXiv:math/0104012. · Zbl 1273.18009
[6] T. LEINSTER, Is there an odd-order group whose order is the sum of the orders of the proper normal subgroups?, Math. Overflow, http://mathoverflow.net/ questions/54851.
[7] W. A. STEIN et. al., Sage Mathematics Software (Version 4.6.1), The Sage Development Team, 2011, http://www.sagemath.org.
[8] W. G. STANTON - J. A. HOLDENER, Abundancy outlaws of the form (s(N) \? t)=N, J. Integer Sequences 10 (2007), Article 09.7.6. · Zbl 1174.11005 · emis:journals/JIS/VOL10/Holdener/holdener7.html · eudml:55989
[9] http://java.ugent.be/\(tdemedts/leinster.\)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.