Canadian Mathematical Bulletin ( IF 0.6 ) Pub Date : 2021-01-26 , DOI: 10.4153/s0008439521000047 Morteza Baniasad Azad 1 , Behrooz Khosravi 2
Let G be a finite group and $\psi (G) = \sum _{g \in G} o(g)$ , where $o(g)$ denotes the order of $g \in G$ . There are many results on the influence of this function on the structure of a finite group G.
In this paper, as the main result, we answer a conjecture of Tărnăuceanu. In fact, we prove that if G is a group of order n and $\psi (G)>31\psi (C_n)/77$ , where $C_n$ is the cyclic group of order n, then G is supersolvable. Also, we prove that if G is not a supersolvable group of order n and $\psi (G) = 31\psi (C_n)/77$ , then $G\cong A_4 \times C_m$ , where $(m, 6)=1$ .
Finally, Herzog et al. in (2018, J. Algebra, 511, 215–226) posed the following conjecture: If $H\leq G$ , then $\psi (G) \unicode[stix]{x02A7D} \psi (H) |G:H|^2$ . By an example, we show that this conjecture is not satisfied in general.
中文翻译:
关于元素阶和的两个猜想
令G为有限群, $\psi (G) = \sum _{g \in G} o(g)$ ,其中 $o(g)$ 表示 $g \in G$ 的阶。关于这个函数对有限群G结构的影响有很多结果。
在本文中,作为主要结果,我们回答了 Tărnăuceanu 的猜想。事实上,我们证明如果G是n阶群且 $\psi (G)>31\psi (C_n)/77$ ,其中 $C_n$ 是n阶循环群,则G是超可解的。此外,我们证明如果G不是n阶超可解群且 $\psi (G) = 31\psi (C_n)/77$ ,则 $G\cong A_4 \times C_m$ ,其中 $(m, 6 )=1$ 。
最后,赫尔佐格等人。在 (2018, J. Algebra , 511, 215–226) 中提出了以下猜想:如果 $H\leq G$ ,则 $\psi (G) \unicode[stix]{x02A7D} \psi (H) |G: H|^2$ 。通过一个例子,我们表明这个猜想一般是不满足的。