Finite Groups with the Set of the Number of Subgroups of Possible Order Containing Exactly Two Elements

Proceedings Mathematical Sciences

Yanheng Chen¹  Guiyun Chen² 

[1] School of Mathematics and Statistics, Southwest University, Chongqing 00, People’s Republic of China$$;School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 000, People’s Republic of China$$

关键词: Finite group;    the number of subgroups of possible order.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let 𝐺 be a finite group, and $n(G)$ be the set of the number of subgroups of possible order of 𝐺. We investigate the structure of 𝐺 satisfying that $n(G) = {1, m}$ for any positive integer 𝑚 > 1. At first, we prove that the nilpotent length of 𝐺 is less than 2. Secondly, we investigate nilpotent groups with $m = p + 1$ or $p^2 + p + 1$ (𝑝 is a prime), and we get the classification of such kinds of groups. At last, we investigate non-nilpotent groups with $m = p + 1$ and get the classification of the groups under consideration.

【 授权许可】

Unknown