# On Two Conjectures about the Sum of Element Orders

## TL;DR

This paper proves a conjecture linking the sum of element orders to supersolvability in finite groups and provides a counterexample to a related conjecture about subgroup sums.

## Contribution

It confirms a conjecture by Trnuceanu connecting the sum of element orders with supersolvability and demonstrates that another proposed inequality does not hold universally.

## Key findings

- Proved that if mbda(G) exceeds a certain bound, G is supersolvable.
- Provided a counterexample to the conjecture about mbda(G) for subgroups.
- Established limitations of the conjectured inequality in group theory.

## Abstract

Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$ denotes the order of $g \in G$. First, we prove that if $G$ is a group of order $n$ and $\psi(G) >31\psi(C_n)/77$, where $C_n$ is the cyclic group of order $n$, then $G$ is supersolvable. This proves a conjecture of M.~{T\u{a}rn\u{a}uceanu}. Moreover, M. Herzog, P. Longobardi and M. Maj put forward the following conjecture: If $H\leq G$, then $\psi(G) \leqslant \psi(H) |G:H|^2$. In the sequel, by an example we show that this conjecture is not satisfied in general.