Let $G$ be a finite group and let $\operatorname{cd}(G)$ be the set of degrees of irreducible characters of $G$. It is known that if for any $m,n \in \operatorname{cd}(G) \setminus \{1\}$, there exists a prime $p$ such that $p \mid m,n$, then $G$ is solvable. Does the converse hold? Thanks in advance.
2026-03-28 20:08:14.1774728494
If $G$ is solvable, is it true that for any $m,n\in\operatorname{cd}(G)$, there exists a prime $p$ such that $p\mid m,n$?
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No, take for example $S_4$, which has character degrees $1,2,$ and $3$.