A character-theoretic criterion for the existence of normal complements to subgroups of finite groups
A subgroup H of a finite group G is said to have the CR property in G if each irreducible complex character of H is the restriction of a character of G. \textit{H. Schmidt} and \textit{F. Richen} [J. Lond. Math. Soc., II. Ser. 7, 168-170 (1973; Zbl 0285.20025)] proved that if H has the CR property in a soluble group G and is a Carter subgroup of G then H has a normal complement in G. The authors generalize this to: Theorem. Let \({\mathcal F}\) be a saturated formation, let G be a finite soluble group, and let H be an \({\mathcal F}\)-projector of G. If H has the CR property in G, then H has a normal complement in G. This result is proved by first studying the properties of a minimal counterexample and then arriving to a contradiction.
scientific article; zbMATH DE number 3955193