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, resolving the remaining case left by Camina, Maróti, Pacifici, Parker, Rekvényi, Saunders, Sotomayor, Tracey and van Beek. For a $p$-regular conjugacy class $B$ of maximal size, their structure results provide an abelian normal $p^{\prime }$-subgroup $M$. We choose a noncentral $r$-element $x\in M$ such that $r\mid |B|$ and $\gcd (|x^G|,pr)=1$, and prove that $C_G(x)/M$ is a $\{p,r\}$-group. For $H=G/C_G(M)$, acting faithfully on $M$ by conjugation, it follows that the stabilizer $H_x$ of $x$ is a Hall $\{p,r\}$-subgroup and that every $p^{\prime }$-element of $H$ has conjugacy-class size whose prime divisors lie in $\{p,r\}$. The theorem of Dolfi and Lucido, together with Burnside's $p^a{q}^b$-theorem, then excludes nonabelian composition factors of $H$.

5 pages