A result on certain sums of element orders in finite groups
Marius Tărnăuceanu
TL;DR
This paper investigates sums of element orders restricted to their $p$-parts, defining $\psi_p(G)=\sum_{x\in G} o(x_p)$ for groups of order $|G|=p^n m$ with $p\nmid m$. It proves the upper bound $\psi_p(G)\le\psi_p(C_{p^nm})$ and characterizes equality precisely as $G\cong H\rtimes C_{p^n}$ with $H$ normal of order $m$, using a partition of $G$ and a Frobenius/Iiyori–Yamaki framework. The paper also extends the result to a general $\psi_{\pi}(G)$ for any set of primes $\pi$ dividing $|G|$, showing $\psi_{\pi}(G)\le\psi_{\pi}(C_n)$ with the equality case giving a semidirect product structure $G\cong H\rtimes C$, where $H$ is the $\pi'$-part and $C$ the $\pi$-part, respectively. These findings provide a structural constraint on finite groups via order-sum invariants and generalize the classical inequality for the total order-sum $\psi(G)$ to $p$-part and prime-set variants.
Abstract
Given a finite group $G$ of order $p^nm$, where $p$ is a prime and $p\nmid m$, we denote by $ψ_p(G)$ the sum of orders of $p$-parts of elements in $G$. In the current note, we prove that $ψ_p(G)\leqψ_p(C_{p^nm})$, where $C_{p^nm}$ is the cyclic group of order $p^nm$, and the equality holds if and only if $G$ is $p$-nilpotent of a particular type. A generalization of this result is also presented.