Let G be a finite group and ψ(G) =∑g∈Go(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″(D2p), where p is a prime number, then G≅Op(G) × Op′(G) and Op(G) is cyclic. As the next result, we show that if G is not a p-nilpotent group and ψ″(G) = ψ″(D2p), then G≅D2p.

中文翻译：

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关于元素阶数之和的猜想的答案

让G是一个有限群并且ψ(G) =∑G∈G○(G)， 在哪里○(G)表示顺序G. 功能ψ''(G) = ψ(G)/|G|2由 Tărnăuceanu 介绍。在 [M. Tărnăuceanu，通过元素阶数之和检测有限群的结构性质，以色列 J. 数学。(2020), doi:10.1007/s11856-020-2033-9]，一些下限ψ”(G)被确定为如果ψ”(G)大于它们中的每一个，那么G是循环的、阿贝尔的、幂零的、超可解和可解的。此外，还有一个关于有限群的开放问题G这样ψ”(G)等于每个下限的数量。在本文中，我们给出了相等条件的答案，这是对 Tărnăuceanu 提出的开放问题的部分答案。此外，在 [M. Baniasad Azad 和 B. Khosravi，通过元素阶数之和确定 p 幂零性和 p 闭合性的标准，交流。代数(2020), doi:10.1080/00927872.2020.1788571]，表明：如果ψ”(G) > ψ”(D2p)， 在哪里p是素数，那么G≅○p(G) × ○p'(G)和○p(G)是循环的。作为下一个结果，我们证明如果G不是一个p- 幂零群和ψ”(G) = ψ”(D2p)， 然后G≅D2p.