On two conjectures about the sum of element orders

Azad, Morteza Baniasad ; Khosravi, Behrooz

Canadian mathematical bulletin, Tome 65 (2022) no. 1, pp. 30-38

Résumé

Let G be a finite group and $\psi (G) = \sum _{g \in G} o(g)$, where $o(g)$ denotes the order of $g \in G$. There are many results on the influence of this function on the structure of a finite group G.In this paper, as the main result, we answer a conjecture of Tărnăuceanu. In fact, 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. Also, we prove that if G is not a supersolvable group of order n and $\psi (G) = 31\psi (C_n)/77$, then $G\cong A_4 \times C_m$, where $(m, 6)=1$.Finally, Herzog et al. in (2018, J. Algebra, 511, 215–226) posed the following conjecture: If $H\leq G$, then $\psi (G) \unicode[stix]{x02A7D} \psi (H) |G:H|^2$. By an example, we show that this conjecture is not satisfied in general.

Détail

DOI : 10.4153/S0008439521000047

Mots-clés : Sum of element orders, supersolvable group, element orders