The influence of order and conjugacy class length on the structure of finite groups (Q1709219)

The authors prove the following theorem: Let $G$ be a finite group with $|G| =|C_n(2)|$ for some $n$ such that $2^n+1= p >5$ is a prime. Assume that there is $g \in G$, $o(g) = p$ and $C_G(g) = \langle g \rangle$. Then $G \cong C_n(2)$. As for most of these theorems, the proof uses the Kegel-Williams prime graph to reduce the problem to almost simple groups, and then the classification of finite simple groups. As the order of $G$ is known, an application of Zsigmondy's theorem reduces this to check about 25 cases.