A note on the converse of the Clifford's theorem and some consequences (Q1091463)

The author proves the following converse of Clifford's theorem: Th. B. Let \(N<G\), G be finite, and for every \(\chi\in Irr(G)\), \(\chi_ N=e(\theta^{g_ 1}+...+\theta^{g_ n})\) where \(g_ i\in N_ G(N)\), \(\theta\in Irr(N)\). Then \(N\trianglelefteq G.\) (We give a shorter proof: Since by condition and reciprocity \(((1_ N)^ G)_ N=a1_ N\) we have \(N\leq \ker (1_ N)^ G\). Since the opposite inclusion is true also, \(N\trianglelefteq G.)\) As a corollary the author proves Th. A. Let \(P\in Syl_ p(G)\). Then \(G=P\times K\Leftrightarrow \chi_ P=n\theta\) for every \(\chi\in Irr(G)\), where \(\theta =\theta (\chi)\in Irr(P)\) and \(n\in {\mathbb{N}}.\) We note that this result is also true for P a Hall subgroup of G (it is needed a new proof by using Brauer's matrix lemma).