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An answer to a conjecture on the sum of element orders
Journal of Algebra and Its Applications ( IF 0.5 ) Pub Date : 2020-12-28 , DOI: 10.1142/s0219498822500670
Morteza Baniasad Azad 1 , Behrooz Khosravi 1 , Morteza Jafarpour 2
Affiliation  

Let G be a finite group and ψ(G) =gGo(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 GOp(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 GD2p.

中文翻译:

关于元素阶数之和的猜想的答案

G是一个有限群并且ψ(G) =GG(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Gp(G) × p'(G)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GD2p.
更新日期:2020-12-28
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