Group Structure from Subgroup and Cyclic Subgroup Counts
Presidency University（总统大学）；Indian Institute of Technology, Jammu（印度技术学院贾姆分校）；Universidad de los Andes（洛斯安第斯大学）；Indian Institute of Science Education and Research, Berhampur（印度科学教育研究学院贝尔哈姆普尔分校）
Angsuman Das, Hiranya Kishore Dey, Cesar Galindo, Khyati Sharma
For a finite group \(G\), let \(\sub(G)\) be the number of subgroups of \(G\), let \(\cyc(G)\) be the number of cyclic subgroups, and let \(π(G)\) be the number of distinct prime divisors of \(|G|\). We study the normalized counts \(λ(G)=\sub(G)/2^{π(G)}\) and \(η(G)=\cyc(G)/2^{π(G)}\). We prove that \(η(G)<5/4\) or \(λ(G)<3/2\) implies that \(G\) is cyclic of squarefree order. The inequalities \(η(G)<2\) and \(λ(G)<5/2\) each force all Sylow subgroups to be cyclic, and hence imply metacyclicity. For the subgroup count, we give an exact arithmetic criterion in the parameters of the corresponding \(ZM\)-presentation. We determine all values with \(1<η(G)<2\) and \(1<λ(G)<5/2\), and prove that \(λ(G)<59/8\) or \(η(G)<4\) implies solvability. Both solvability bounds are sharp. We also describe how cyclic direct factors of coprime order affect the two normalized counts.
设$G$是一个群。\textit{The permutability graph of cyclic subgroups of $G$}（记作$Γ_c(G)$）是以$G$的所有真循环子群为顶点的图，且$Γ_c(G)$中的两个不同顶点相邻当且仅当相应的子群在$G$中置换.本文对循环子群的置换图属于以下任一种的有限群进行了分类：二部群、树群、星图群、无三角形群、完全二部群、$P_n$群、$C_n$群、$K_4$群、无$K_{1,3}$群、单循环群。对循环子群置换图为平面的阿贝尔群进行了分类。同时，我们还研究了这些图的连通性、直径、围长、完全不连通性、完备性和正则性。
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