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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.
20C15 Ordinary representations and characters