The solvability of a finite group by the sum of powers of element orders

TL;DR

This paper introduces a new criterion based on the sum of powers of element orders to determine the solvability of finite groups, providing insights beyond previous methods.

Contribution

It presents a novel criterion involving the function _k(G) for assessing group solvability, extending the understanding of group structure analysis.

Findings

New solvability criterion using sum of powers of element orders

Applicable to groups not solvable by earlier criteria

Highlights the usefulness of _k(G) for k>1

Abstract

We prove a new criterion for the solvability of the finite groups, depending on the function $\psi_k(G)$ which is defined as the sum of $k$-th powers of the element orders of $G$. We show that our result can be used to show the solvability of some groups for which the solvability does not follow from earlier similar kind of results and we emphasize the following: looking at $\psi_k(G)$ for $k>1$ can be useful to get further pieces of information about the group $G$.