A note on $\pi$-partial characters of $\pi$-separable groups

Abstract

Let $\pi$ be a set of prime numbers and G be a $\pi$-separable group. If $\varphi (1)_{\pi '}^{2}$ divides $|G: \ker \varphi |_{\pi '}$ for every $\pi$-partial character $\varphi \in \mathrm{I}_{\pi }(G)$, then G has a normal Hall $\pi '$-subgroup, where $\varphi (1)_{\pi '}$ denotes the $\pi '$-part of $\varphi (1)$ and $\mathrm{I}_{\pi }(G)$ is the set of irreducible $\pi$-partial characters of G.