1. Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements
- Creator:
- Shao, Changguo and Jiang, Qinhui
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- finite group and number of subgroups of possible orders
- Language:
- English
- Description:
- Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups $G$ whose set of numbers of subgroups of possible orders $n(G)$ has exactly two elements. We show that if $G$ is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then $G$ has a normal Sylow subgroup of prime order and $G $ is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with $n(G)=\{1, q+1\}$ for some prime $q$. In particular, $G$ is supersolvable under this condition.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public