Simple \(\pmb {\mathscr {B}}_{\psi }\)-groups

In a finite group G, \( \psi (G) \) $\psi (G)$ denotes the sum of element orders of G. A finite group G is said to be a \(\mathscr {B}_{\psi }\) ${\mathcal{B}}_{\psi }$ -group if \( \psi (H) < |G| \) $\psi (H)<|G|$ for any proper subgroup H of G. In Lazorec (Rev Real Acad Cienc Exact Fis Nat Ser A Mat 117:448, 2023), Lazorec put forward the following question: Question. What can be said about the \(\mathscr {B}_{\psi }\) ${\mathcal{B}}_{\psi }$ property of the finite simple groups \( {\text {PSL}}(2, q) \) $\text{PSL}(2,q)$ ? In this paper, we show that if S is a finite simple group, such that \( S \ne Alt(n) \) $S\ne Alt(n)$ for any \( n \ge 14 \) $n\ge 14$ , then S is a \(\mathscr {B}_{\psi }\) ${\mathcal{B}}_{\psi }$ -group.