An answer to a conjecture on the sum of element orders.

Let G be a finite group and ψ (G) = ∑ g ∈ G o (g) , where o (g) denotes the order of g. The function ψ ′ ′ (G) = ψ (G) / | G | 2 was introduced by Tărnăuceanu. In [M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. (2020), doi:10.1007/s11856-020-2033-9], some lower bounds for ψ ″ (G) are determined such that if ψ ″ (G) is greater than each of them, then G is cyclic, abelian, nilpotent, supersolvable and solvable. Also, an open problem aroused about finite groups G such that ψ ″ (G) is equal to the amount of each lower bound. In this paper, we give an answer to the equality condition which is a partial answer to the open problem posed by Tărnăuceanu. Also, in [M. Baniasad Azad and B. Khosravi, A criterion for p-nilpotency and p-closedness by the sum of element orders, Commun. Algebra (2020), doi:10.1080/00927872.2020.1788571], it is shown that: If ψ ″ (G) > ψ ″ (D 2 p) , where p is a prime number, then G ≅ O p (G) × O p ′ (G) and O p (G) is cyclic. As the next result, we show that if G is not a p -nilpotent group and ψ ″ (G) = ψ ″ (D 2 p) , then G ≅ D 2 p . [ABSTRACT FROM AUTHOR]