# Characterization of P-groups By Sum of Element Orders

## TL;DR

This paper characterizes certain finite p-groups using the sum of element orders function, providing new insights into their structure based on this numerical invariant.

## Contribution

It introduces a novel characterization of finite p-groups through the sum of element orders, linking group structure to this specific numerical property.

## Key findings

- Identifies conditions under which p-groups are characterized by their sum of element orders
- Provides classifications of p-groups based on the value of ψ(G)
- Establishes relationships between group order and sum of element orders

## Abstract

Let $G$ be a finite group. Then we denote $\psi(G) = \sum_{x\in G}o(x)$ where $o(x)$ is the order of the element $x$ in $G$. In this paper we characterize some finite $p$-groups ($p$ a prime) by $\psi$ and their orders.