Subgroups with the character restriction property
A subgroup H of a finite group G satisfies the CR (character restriction) property in G if each ordinary irreducible character of H is the restriction of one of G. It is sufficient for this that H have a normal complement in G, but not necessary. In this paper, some necessary conditions for the CR property are given. First, if \(\pi\) is a set of primes, H is maximal among solvable \(\pi\)- subgroups of G and the CR property holds, then H is a Hall \(\pi\)-subgroup of G, and G has a normal \(\pi\)-complement. Second, if H is the normalizer of a Sylow subgroup and satisfies the CR property in G, then H has a normal complement in G. Third, if G is solvable, \({\mathcal F}\) is a saturated formation, and an \({\mathcal F}\)-normalizer of H in G satisfies the CR property, then H has a normal complement in G. Along the way, a variation on the difficult Orbit Theorem of \textit{T. O. Hawkes} and \textit{G. R. Jones} [J. Algebra 94, 364-381 (1985)] is obtained.
scientific article; zbMATH DE number 3944123