Character restriction and relative normal complements (Q1115520)

A subgroup H of a finite group G is said to be CR if every complex irreducible character of H is the restriction of a character of G. An obvious sufficient condition for CR is having a normal complement. The main theorem is that the converse is true for Hall subgroups. This has the consequence that if H is a subgroup of G, \(H_ 0\) is a normal subgroup of H and \((| G:H|,| H:H_ 0|)=1\), then \(H/H_ 0\) has a relative normal complement in G if and only if every character of \(H/H_ 0\) is the restriction of a character of G, since \(HH_ 0^ G/H_ 0^ G\cong H/H_ 0\) satisfies the hypothesis of the main theorem. The proof of this uses the classification and many intricate properties of finite simple groups.