ABSTRACT
We prove several results detecting cyclicity or nilpotency of a finite group G in terms of inequalities involving the orders of the elements of G and the orders of the elements of the cyclic group of order |G|. We prove that, among the groups of the same order, the number of cyclic subgroups is minimal for the cyclic group, and the product of the orders of the elements is maximal for the cyclic group.
2010 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgments
We are really grateful to Andrea Lucchini and Federico Menegazzo for helpful discussions, comments and suggestions. We are also very grateful to the referee for his/her very careful reading of a previous version of the paper.