Solubility from a disconnected common-divisor graph on $p$-regular conjugacy-class sizes

Abstract

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. This settles the remaining case identified by Camina and collaborators. Their structure results provide an abelian normal $p^{\prime }$-subgroup $M$. We choose a noncentral $r$-element $x\in M$, with $r\ne p$, such that $[G:C_G(x)]$ is coprime to $pr$, and show that $C_G(x)/M$ is a $\{p,r\}$-group. Thus $C_G(x)/M$ is a Hall $\{p,r\}$-subgroup of $G/M$. The separation between the two graph components then implies that every $p$-regular conjugacy-class size in $G/M$ has all its prime divisors in $\{p,r\}$. A theorem of Dolfi and Lucido, together with Burnside's $p^a{q}^b$-theorem, yields the solubility of $G/M$ and hence of $G$.

arXiv comment6 pages; proof and exposition substantially revised; main theorem unchanged