## On groups all of whose irreducible characters are quasiprimitive. (Über Gruppen, deren irreduzible Charaktere sämtlich quasiprimitiv sind.)(German)Zbl 0874.20005

A finite group $$G$$ is called quasi-primitive (characteristically quasi-primitive) if, for every irreducible character $$\chi$$ of $$G$$ and every normal (characteristic) subgroup $$N$$ of $$G$$, the restriction $$\chi_N$$ is a multiple of an irreducible character of $$N$$. The authors show that a finite group $$G$$ is quasi-primitive if and only if $$G/Z(G)$$ is a direct product of nonabelian finite simple groups. Their proof involves the classification of finite simple groups. They also show that a characteristically quasi-primitive finite solvable group $$G$$ is nilpotent of class at most 2, with $$G'$$ a direct product of elementary abelian groups of prime power order.

### MSC:

 20C15 Ordinary representations and characters