Hall subgroups and stable Brauer characters (Q2716982)

Let \(G\) be a finite \(\pi\)-separable group and \(H\) a Hall \(\pi\)-subgroup of \(G\). An irreducible Brauer character \(\alpha\) of \(H\) is called \(G\)-stable if \(\alpha(x)=\alpha(y)\) whenever \(x,y\in H\) are \(p\)-regular and \(G\)-conjugate. The author proves that if \(\alpha\) is a \(G\)-stable irreducible Brauer character of \(H\), then \(\alpha\) extends to a Brauer character of \(G\). As a corollary, one has that every irreducible Brauer character of \(H\) extends to a Brauer character of \(G\) if and only if whenever \(x,y\in H\) are \(p\)-regular and \(G\)-conjugate, \(x\) and \(y\) are also \(H\)-conjugate.