Extending characters from Hall subgroups.

Abstract

Suppose that $G$ is a finite $\pi$-separable group. A classical result asserts that all irreducible characters of a Hall $\pi$-subgroup $H$ of $G$ extend to $G$ if and only if $H$ has a normal complement in $G$. Now, we fix a prime $p$ and analyze when only the $p^{\prime }$-degree irreducible characters of $H$ extend to $G$.