Almost and quasi-Leinster groups

Pleşca, Iulia-Cătălina; Tărnăuceanu, Marius

doi:10.2478/auom-2025-0031

Abstract

In this paper, we extend the parallelism between perfect numbers and Leinster groups, by introducing the new concepts of almost and quasi-Leinster groups which parallel almost and quasi-perfect numbers. These are small deviations from perfect numbers; very few results and/or examples are known about them.

We investigate nilpotent almost/quasi-/Leinster groups and find some examples and conditions for the existence of such groups for classes of nonnilpotent groups: ZM (Zassenhaus metacyclic) groups, affine groups, dihedral groups and dicyclic groups.

References

S.J. Baishya, Revisiting the Leinster groups, C.R. Math. 352 (2014), 1-6.
Search in Google Scholar Back to article
S.J. Baishya, On Leinster groups of order pqrs, arXiv:1911.04829.
Search in Google Scholar Back to article
W.C. Calhoun, Counting subgroups of some finite groups, Amer. Math. Monthly 94 (1987), 54-59.
Search in Google Scholar Back to article
K. Conrad, Normal Subgroups of Aff (F), https://kconrad.math.uconn.edu/blurbs/grouptheory/affinenormal.pdf
Search in Google Scholar Back to article
K. Conrad, Dihedral groups ii, http://www.math.uconn.edu/blurbs/grouptheory/dihedral2.pdf
Search in Google Scholar Back to article
R. Guy, Unsolved Problems in Number Theory, third edition, Springer Verlag, 2004.
Search in Google Scholar Back to article
B. Huppert, Endliche Gruppen, I, Springer Verlag, Berlin, 1967.
Search in Google Scholar Back to article
I.M. Isaacs, Finite Group Theory, Amer. Math. Soc., Providence, R.I., 2008.
Search in Google Scholar Back to article
T. Leinster, Perfect numbers and groups, Eureka 55 (2001), 1727.
Search in Google Scholar Back to article
D. A. Lingenbrink Jr., A New Subgroup Chain for the Finite Affine Group, (2014), HMC Senior Theses, 55. https://scholarship.claremont.edu/hmctheses/55
Search in Google Scholar Back to article
T. De Medts and A. Maróti, Perfect numbers and finite groups, Rend. Semin. Mat. Univ. Padova 129 (2013), 1734.
Search in Google Scholar Back to article
G. A. Miller, Form of the number of subgroups of prime power groups, Bull. Amer. Math. Soc. 26 (1919), 6672.
Search in Google Scholar Back to article
J. Nieuwveld, A note on Leinster groups, https://www.math.ru.nl/~bosma/Students/JorisNieuwveld/
Search in Google Scholar Back to article
H. Saydi, Normal supercharacter theories of the dicyclic groups, Int. Electron. J. Algebra, 37 (2025), 125-139.
Search in Google Scholar Back to article
W. R. Scott, Group Theory, Dover Publications, New York, 1987.
Search in Google Scholar Back to article
M. Trnuceanu, The normal subgroup structure of ZM-groups, Ann. Mat. Pura Appl. 193 (2014), 10851088.
Search in Google Scholar Back to article
H. J. Zassenhaus, The theory of groups, Chelsea Publishing Company, 1949.
Search in Google Scholar Back to article
The GAP Group, GAP groups, algorithms, and programming, version 4.11.0, (2020), https://www.gap-system.org
Search in Google Scholar Back to article
The Great Internet Mersenne Prime Search, https://www.mersenne.org/.
Search in Google Scholar Back to article
MathOverflow, Is there an odd-order group whose order is the sum of the orders of the proper normal subgroups?, (2011), https://mathoverflow.net/questions/54851
Search in Google Scholar Back to article

Almost and quasi-Leinster groups

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