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Solubility from a disconnected common-divisor graph on $p$-regular conjugacy-class sizes
Let $G$ be a finite group and let $p$ be a prime. We prove that if the common-divisor graph on the nontrivial conjugacy-class sizes of $p$-regular elements of $G$ is disconnected, then $G$ is soluble, resolving the remaining case left by Camina, Maróti, Pacifici, Parker, Rekvényi, Saunders, Sotomayo
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