On the number of conjugacy classes of $\pi$-elements in finite groups

摘要

Let $G$ be a finite group and $\pi$ be a set of primes. We show that if thenumber of conjugacy classes of $\pi$-elements in $G$ is larger than $5/8$ timesthe $\pi$-part of $|G|$ then $G$ possesses an abelian Hall $\pi$-subgroup whichmeets every conjugacy class of $\pi$-elements in $G$. This extends andgeneralizes a result of W. H. Gustafson.