Extending characters from Hall subgroups. (Q418673)

Let \(G\) be a finite \(\pi\)-separable group. \textit{C.-H. Sah} proved that if \(H\) is a Hall \(\pi\)-subgroup of \(G\) then all irreducible complex characters of \(H\) extend to \(G\) if and only if \(G\) has a normal \(\pi\)-complement [Ill. J. Math. 6, 282-291 (1962; Zbl 0105.25602)].NEWLINENEWLINE In this paper the authors fix a prime \(p\) and analyze when only the \(p'\)-degree irreducible characters of \(H\) extend to \(G\). In particular they prove the following theorem. Theorem. Let \(G\) be a finite \(\pi\)-separable group. Let \(H\) be a Hall \(\pi\)-subgroup of \(G\), let \(K\) be a \(\pi\)-complement of \(G\), and let \(p\) be a prime. Then every \(\chi\in\text{Irr}(H)\) of \(p'\)-degree extends to \(G\) if and only if there is \(P\in\text{Syl}_p(H)\) such that \(N_G(P)\subseteq N_G(K)\).