论文标题
非循环有限组中元素订单的权力之和的确切上限
An exact upper bound for the sum of powers of element orders in non-cyclic finite groups
论文作者
论文摘要
对于有限的$ g $，令$ψ（g）$表示$ g $的元素订单之和。此功能是由Amiri，Amiri和Isaacs在2009年引入的，他们证明，对于任何有限的$ g $ of订单$ n $，$ψ（g）$是最大的，并且仅当$ g \ simeq \ simeq \ simeq \ mathbb {z} _n $中此外，2018年的Herzog，Longobardi和Maj证明，如果$ G $是非循环的，则$ψ（g）\ leq \ frac {7} {11} {11}ψ（\ Mathbb {z} _n）$。 Amiri和Amiri在2014年推出了$ψ_K（g）$，该功能定义为$ g $的$ k $ th元素的总和，它们表明，对于每个正整数$ k $，$ψ_k（g）$的总和也是$ g $ cyclic的最大值。 在本文中，我们已经能够证明，如果$ g $是$ n $订单的非环保组，则$ψ_k（g）\ leq \ frac {1+3.2^k} {1+2.4^k+2^k} k} k}在我们的结果中设置$ k = 1 $，我们立即获得了Herzog等人的结果。作为简单的推论。此外，还可以使用有限的Abelian $ P $ -Groups $ G $获得$ψ_K（g）$的递归公式，使用该公式可以明确地找出有限ABELIAN组$ G $的$ψ_K（g）$的确切值。
For a finite group $G$, let $ψ(G)$ denote the sum of element orders of $G$. This function was introduced by Amiri, Amiri, and Isaacs in 2009 and they proved that for any finite group $G$ of order $n$, $ψ(G)$ is maximum if and only if $G \simeq \mathbb{Z}_n$ where $\mathbb{Z}_n$ denotes the cyclic group of order $n$. Furthermore, Herzog, Longobardi, and Maj in 2018 proved that if $G$ is non-cyclic, $ψ(G) \leq \frac{7}{11} ψ(\mathbb{Z}_n)$. Amiri and Amiri in 2014 introduced the function $ψ_k(G)$ which is defined as the sum of the $k$-th powers of element orders of $G$ and they showed that for every positive integer $k$, $ψ_k(G)$ is also maximum if and only if $G$ is cyclic. In this paper, we have been able to prove that if $G$ is a non-cyclic group of order $n$, then $ψ_k(G) \leq \frac{1+3.2^k}{1+2.4^k+2^k} ψ_k(\mathbb{Z}_n)$. Setting $k=1$ in our result, we immediately get the result of Herzog et al. as a simple corollary. Besides, a recursive formula for $ψ_k(G)$ is also obtained for finite abelian $p$-groups $G$, using which one can explicitly find out the exact value of $ψ_k(G)$ for finite abelian groups $G$.
