Almost and quasi Leinster groups
Iulia-Cătălina Pleşca, Marius Tărnăuceanu
TL;DR
This work extends the Leinster-group framework by defining almost Leinster ($D(G)=2|G|-1$) and quasi-Leinster ($D(G)=2|G|+1$) groups and analyzes their existence across key finite group classes. It proves that nilpotent almost/quasi-Leinster groups are precisely cyclic with almost/quasi-perfect orders, and then treats non-nilpotent families (ZM-groups, affine groups, dihedral and dicyclic groups) to identify where such groups can exist, providing explicit examples and structural constraints. Notably, quasi-Leinster ZM-groups exist with specific triples and affine almost-Leinster groups arise when $q-1$ is perfect; dihedral and dicyclic cases are tightly constrained, yielding few instances such as $\mathrm{Dic}_{12}$ being Leinster. The results extend Leinster-group theory by drawing deep parallels with almost and quasi-perfect numbers and charting concrete classifications within several standard finite-group families.
Abstract
In this paper, we study the parallelism between perfect numbers and Leinster groups and continue it 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 non-nilpotent groups: ZM (Zassenhaus metacyclic) groups, dihedral generalised groups, generalised dyciclic groups and affine groups.