On the character theory of fully ramified sections
The following theorem is a consequence of a stronger result by the author [in Am. J. Math. 95, 594-635 (1973; Zbl 0277.20008)] whose proof requires some lengthy and relatively difficult arguments: Let L and K be normal subgroups of the group G such that \(L\leq K\) and K/L is abelian of odd order. Let \(\phi \in Irr(L)\) be invariant in G and assume \(\phi^ K=e\theta\) for some \(\theta \in Irr(K)\) and some integer e. Then there exists a subgroup U of G such that (a) \(UK=G\) and \(U\cap K=L;\) (b) \(| Irr(G| \phi)| =| Irr(U| \phi)|\); (c) if \(| K/L|>1\) and \(\xi \in Irr(U| \phi)\) then \(\xi^ G\) is reducible; (d) if \(| K/L|>1\) and \(\chi \in Irr(G| \phi)\) then \(\chi_ U\) is reducible. The purpose of this paper is to present a simple direct proof. A purely group-theoretic result is used to produce the subgroup U. Among other applications to the character theory of solvable groups the following corollary is given: Let G be a solvable group of odd order, and let A be a group acting on G such that \((| A|,| G|)=1\). Then the number of A-fixed irreducible characters of G is equal to the total number of irreducible characters of \(C_ G(A)\).
scientific article; zbMATH DE number 3842109