Let \(H_ 0\triangleleft H<G\), G be a finite group; \(\bar H=H/H_ 0\), and let the triple \((G,H,H_ 0)\) satisfy (B) If \(x,y\in H\) are fused in G then \(\bar x,\bar y\) are conjugate in \(\bar H.\)- Then \((G,H,H_ 0)\) is called a B-triple.
Let \(\pi =\pi (\bar H)\) be the set of all prime divisors of \(| \bar H|\). A B-triple \((G,H,H_ 0)\) is called \(B_{\pi}\)-triple if \(\cup_{x\in G}H^ x\) contains all \(\pi\)-elements of G. For every \(x\in G\) we set \(x=x_{\pi}x_{\pi '}=x_{\pi '}x_{\pi}\) where \(x_{\pi}\) is the \(\pi\)-part of x. Let \(H_ 0,H_ 1,...,H_ n\) be the full images in H of all \(\bar H-\)classes. Let \(G_ i=\{g\in G|\) \(g_{\pi}\sim x_{\pi}\) with \(x\in H_ i\}\). Then \(G=\cup^{n}_{i=0}G_ i\) is a disjoint union. We set for every \(\phi\in Irr(\bar H):\) \(\phi^*(G_ i)=\phi (H_ i)\), \(i\in \{0,1,...,n\}.\)
Theorem 1. Let \((G,H,H_ 0)\) be a \(B_{\pi}\)-triple, \(\pi =\pi (\bar H)\), \(\bar H=H/H_ 0\). If \(\phi^*\) is a generalized character for all \(\phi\in Irr(\bar H)\), then \(\phi^*\in Irr(G)\), \(G_ 0\triangleleft G\), \(G=HG_ 0\), \(H\cap G_ 0=H_ 0\). As a corollary we obtain the well-known result of E. C. Dade [J. Aust. Math. Soc. 19, 257-262 (1975; Zbl 0319.20031)]. If \(H_ 0=1\) then Th. 1 implies a theorem of Brauer-Suzuki [see Th. 8.22 in I. M. Isaacs, Character theory of finite groups (1976; Zbl 0337.20005)]. If for \(i>0\), \(G_ i=\cup_{x\in G}H^ x_ i\), \(G_ 0=G-\cup^{n}_{i=0}G_ i\), \((G,H,H_ 0)\) a B- triple. Then \(\phi^*\) is a generalized character for all \(\phi\in Irr(\bar H)\Leftrightarrow H\cap H^ x\leq H_ 0\) for all \(x\in G-H\) (Th.4).