An exact upper bound for the sum of powers of element orders in non-cyclic finite groups

2022/08/10 by Hiranya Kishore Dey, Dey, Hiranya Kishore, Archita Mondal +1 · 1 citation
Computer Science · Mathematics · #20D60 #20E34 #20F18 #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2208.05161

openalex publication_date 2022/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a finite group G, let ψ(G) denote the sum of element orders of G. This function was introduced by Amiri, Amiri, and Isaacs in 2009 and they proved that for any finite group G of order n, ψ(G) is maximum if and only if G ≃ ℤ_n where ℤ_n denotes the cyclic group of order n. Furthermore, Herzog, Longobardi, and Maj in 2018 proved that if G is non-cyclic, ψ(G) ≤ (7)/(11) ψ(ℤ_n). Amiri and Amiri in 2014 introduced the function ψ_k(G) which is defined as the sum of the k-th powers of element orders of G and they showed that for every positive integer k, ψ_k(G) is also maximum if and only if G is cyclic. In this paper, we have been able to prove that if G is a non-cyclic group of order n, then ψ_k(G) ≤ (1+3.2^k)/(1+2.4^k+2^k) ψ_k(ℤ_n). Setting k=1 in our result, we immediately get the result of Herzog et al. as a simple corollary. Besides, a recursive formula for ψ_k(G) is also obtained for finite abelian p-groups G, using which one can explicitly find out the exact value of ψ_k(G) for finite abelian groups G.

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• On the Sum of Element Orders in Finite Abelian Groups

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