Given a finite group $G$ and $x\in G$, the class size of $x$ in $G$ is called odd-square-free if it is not divisible by the square of any odd prime number. In this paper, we show that if $G$ is a nonsolvable finite group, all of whose class sizes are odd-square-free, then we have some control on the structure of $G$, which is an answer to the dual of the question mentioned by Huppert in .
ROBATI, SAJJAD MAHMOOD and BALAMAN, ROGHAYEH HAFEZIEH
"Non-solvable groups all of whose indices are odd-square-free,"
Turkish Journal of Mathematics: Vol. 46:
3, Article 5.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss3/5