Inequalities detecting structural properties of a finite group

2015/03/01 by Martino Garonzi, Garonzi, Martino, Massimiliano Patassini +1 · 3 citations
Engineering · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1503.00355

openalex publication_date 2015/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove several results detecting ciclicity 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.

Cited by

• The influence of the prime omega function on the product of element orders in finite groups
• Recognition by the set of exponents in the prime factorization of the product of element orders
• On the global breadth of finite groups with nontrivial partitions

Related

• An inequality detecting nilpotency of finite groups
• A criterion for nilpotency of a finite group by the sum of element orders
• Detecting structural properties of finite groups by the sum of element orders
• On the structure of finite groups determined by the arithmetic and geometric means of element orders
• A nilpotency criterion for finite groups