Inner & Outer P-Group

Theorem

Let $G$ be a finite inner-cyclic group，then $G$ is isomorphic to one of the following three types of groups：

• $G\cong {\mathbb{Z}}_p\times {\mathbb{Z}}_p$;
• $G$ is the quaternion group of order $8$;
• $G=⟨a,b⟩$ ，with the following defining relations:

$$a^p=1,b^{q^m}=1,b^{-1}ab=a^r$$

where $p,q$ are distinct primes, $m,r$ are positive integers, and $r\not\equiv 1(\mathrm{mod}p),r^q\equiv 1(\mathrm{mod}p)$.

• 内幂零群