On the sum of element orders in finite abelian groups
Abstract
Abstract Let $$\psi (G) = \sum _{g \in G} o(g)$$ 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 $$C_n$$ C n maximizes $$\psi $$ 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, $$\psi (G) = \psi (H)$$ ψ ( G ) = ψ ( H ) if and only if G and H have the same order type.
Fields of Study & Subjects
Finite Group Theory Research
Rings
Modules
and Algebras
Limits and Structures in Graph Theory
Publication details
- Journal / source
- Journal of Algebraic Combinatorics
- Publisher
- Springer Science and Business Media LLC
- Volume
- 63
- Issue
- 4
- Publication date
- 1 June 2026
- Language
- EN