On conjugacy class size of finite groups.
Suppose $G$ is a finite group such that the set of all the conjugacy class size is $\{1,2,\dots,n\}$, where $n$ is a natural number. Is it true that $n\leq 3$? Thanks in advance.
Solution 1:
Yes, $n$ should be $\leq 3$. It is proved by Bianchi, Gillio and Casolo (see here) that in a finite group $G$ such that the two largest (non-central) conjugacy class sizes $n < m$ are coprimes, then any conjugacy class in $G$ has size $1$, $n$ or $m$.