On the Sum of Element Orders in Finite Abelian Groups

2025/11/11 by Amiri, Mohsen
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2511.08320

Abstract

Let ψ(G) = ∑_{g ∈ G} o(g) denote the sum of element orders of a finite group G. It is known that among groups of order n, the cyclic group C_n maximizes ψ. Tărnăuceanu proved that two finite abelian p-groups of the same order are isomorphic if and only if they have the same sum of element orders, and conjectured this for arbitrary finite abelian groups. In this paper, we confirm the conjecture by proving a stronger result: for finite LCM-groups G and H of the same order, ψ(G) = ψ(H) if and only if G and H are the same order type.

Citations

• On a bijection between a finite group to a non-cyclic group with divisibility of element orders
• On the structure of LC-nilpotent groups
• An exact upper bound for the sum of powers of element orders in non-cyclic finite groups
• The order of the product of two elements in the periodic groups
• A bijection from a finite group to the cyclic group with a divisibility property on the element orders
• Detecting structural properties of finite groups by the sum of element orders
• The second maximal groups with respect to the sum of element orders
• A Criterion for Solvability of a Finite Group by the Sum of Element\n Orders

Related

• A generalization of a result on the sum of element orders of a finite\n group
• An exact upper bound for sums of element orders in non-cyclic finite groups
• A result on certain sums of element orders in finite groups
• An exact upper bound for the sum of powers of element orders in non-cyclic finite groups
• The second maximal groups with respect to the sum of element orders