The minimum sum of element orders of finite groups

‎Let $ G $ be a finite group and \( \psi(G)=\sum_{g\in G}o(g) \)‎, ‎where $ o(g) $ denotes the order of $g\in G$‎. ‎We show that the Conjecture ۴.۶.۵ posed in [Group Theory and Computation‎, ‎(۲۰۱۸) ۵۹-۹۰]‎, ‎is incorrect‎. ‎In fact‎, ‎we find a pair of finite groups $G$ and $S$ of the same order such that $ \psi(G)<\psi(S)$‎, ‎with $G$ solvable and $S$ simple‎.

Keywords:

‎Finite group‎ , ‎element orders sum‎ , ‎solvable group‎ , ‎simple group