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On large proportions of cyclic subgroups in finite group | ||
| International Journal of Group Theory | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 04 آبان 1405 | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2026.148673.2019 | ||
| نویسندگان | ||
| Xiaofang Gao1؛ Rulin Shen* 2 | ||
| 1School of Mathematics and Statistics, Hubei University | ||
| 2Department of Mathmetics, Hubei Minzu University | ||
| چکیده | ||
| For a finite group $G$, let $c(G)$ denote the number of cyclic subgroups of $G$ and define the function $\alpha(G) = c(G)/|G|$, which measures the proportion of cyclic subgroups in $G$. In this paper, we establish a complete classification of finite non $2$-groups satisfying $17/24 \leq \alpha(G) < 3/4$, extending previous work on the distribution of cyclic subgroups in finite groups. We prove that any such group $G$ is isomorphic to a direct product $Z_2^n \times X$ for some non-negative integer $n$, where $X$ is one of the following six groups: the direct products $S_3 \times S_3$ and $S_3 \times D_8$, the semidirect products $\mathrm{SmallGroup}(24,8) \cong Z_3 \rtimes D_8$, $\mathrm{SmallGroup}(72,33) \cong Z^2_{3} \rtimes D_8$, $\mathrm{SmallGroup}(96,81) \cong Z_4 \rtimes D_{24}$, or the symmetric group $S_4$. The corresponding values of $\alpha(X)$ are $13/18$, $35/48$, and $17/24$ for the latter four groups, respectively. This classification completely determines the structure of all finite non $2$-groups with cyclic subgroup proportions in the specified interval $[17/24, 3/4)$. | ||
| کلیدواژه‌ها | ||
| Supersolvable groups؛ Number of cyclic subgroups؛ Non $2$-groups | ||
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